SÁCH HDSD SPSS: Understanding statistics for the social sciences with ibm® spss® (2018)

Understanding Statistics for the Social Sciences with IBM SPSS (2018) là một cuốn sách giới thiệu về cách sử dụng phần mềm SPSS để phân tích thống kê trong các ngành khoa học xã hội. Cuốn sách này cung cấp các lời giải thích rõ ràng về các khái niệm thống kê cơ bản và giới thiệu chương trình IBM SPSS để minh họa cách thực hiện phân tích thống kê thông qua các phương pháp pointandclick và syntax file phổ biến. Điểm nhấn của cuốn sách này là chỉ cho sinh viên thấy việc phân tích dữ liệu bằng SPSS rất dễ dàng một khi họ đã học được những điều cơ bản. Cuốn sách này cung cấp một lời giải thích rõ ràng về mục đích của các thủ tục thống kê cụ thể và tránh sử dụng phương pháp nấu ăn thông thường, điều này ít đóng góp cho việc hiểu biết của sinh viên về lý do tại sao kết quả chính xác được đạt được. Lợi ích của việc học chương trình phần mềm IBM SPSS ở cấp độ lớp học đầu tiên là rằng hầu hết sinh viên khoa học xã hội sẽ sử dụng chương trình này trong những năm sau của họ. Điều này là do SPSS là một trong những gói thống kê phổ biến nhất hiện có. Việc học cách sử dụng chương trình này ngay từ đầu không chỉ giúp sinh viên làm quen với tính tiện ích của chương trình mà còn cung cấp cho họ kinh nghiệm để sử dụng chương trình để thực hiện các phân tích phức tạp hơn trong những năm sau này.

Introduction to the Scientific Methodology of Research

Introduction

Many students wonder why research is essential in psychology, questioning the relevance of scientific investigation and statistical analysis to understanding human behavior The fundamental reason is that without objective, empirical verification through scientific methods, the progress of psychology as a science would cease, leading to stagnation in knowledge, technology, and innovation Human evolution, growth, and progress rely on our innate desire to learn, understand, and scientifically explain phenomena such as why A causes B or why people behave in certain ways Mastering scientific research methods is crucial for psychologists to explore, explain, and advance our understanding of human behavior effectively.

The Scientific Approach versus the Layperson’s Approach to

A scientific approach to knowledge differs significantly from a layperson’s perspective, which is often subjective and rooted in intuition and everyday observations In contrast, scientific knowledge is built on systematic observation and direct experimentation For instance, if a friend claims that men are smarter than women based on personal observations, a budding scientist would question the basis of this conclusion, asking about the number of men and women observed This highlights the importance of rigorous, evidence-based methods in scientific inquiry.

Determining whether men and women truly represent their respective populations requires more than anecdotal evidence; it demands empirical proof The scientific approach to understanding intelligence relies on objective methodology rather than speculation or guesswork This involves using rigorous scientific techniques such as random sampling, selecting appropriate research designs, and applying probability theory and hypothesis testing Statistics plays a vital role in analyzing data and enabling researchers to draw valid conclusions about psychological phenomena like intelligence, ensuring findings are based on solid evidence rather than subjective judgment.

Before examining the role that statistics plays in scientific research, let’s review how these three methodological ‘pillars’ provide the foundation for the scientific investigation of social/psychological phenomena.

Sampling is crucial in social science research as it allows researchers to generalize findings from a subset to the entire population, making studies feasible when investigating large groups like Bangkok's 10 million residents Since studying the whole population is often impractical, researchers rely on carefully selected samples, with probability (random) sampling ensuring that every individual has an equal chance of selection This approach guarantees that the results obtained from the sample accurately reflect the characteristics of the larger population, enabling valid generalizations.

In quantitative research, the most common study designs are broadly categorized into between-groups design and correlational design, each serving different research objectives Between-groups design involves comparing different groups to examine causal relationships or differences, while correlational design focuses on exploring the relationships and associations between variables without implying causation These designs vary primarily based on the study's aims and the specific research questions being addressed Understanding these fundamental differences helps researchers select the appropriate approach for their scientific investigation.

When a researcher is interested in investigating whether the manipulation of an independent variable (IV) has some effect on the dependent variable

(DV), then the between-groups design is appropriate The between-groups design can be further classified into the univariate approach and the multivariate approach.

The Univariate Approach

A univariate approach focuses on analyzing a single dependent variable (DV) within the research design For instance, when examining gender differences (the independent variable, IV) in problem-solving skills (the DV) using one measurement of problem-solving ability, the study adopts a univariate design This method allows researchers to explore specific relationships and differences related to one key outcome, such as problem-solving scores, as illustrated in Table 1.1 with data from a sample.

A univariate design can include multiple independent variables, forming a factorial design when more than one IV is involved For instance, to examine the effects of gender and age on problem-solving skills, researchers can classify age into two categories—young and old—and investigate both variables simultaneously This 2 × 2 factorial design results in four groups: male-young, male-old, female-young, and female-old, allowing researchers to analyze the joint effects of gender and age on the dependent variable For example, data collected from these groups can reveal differences in problem-solving scores, as shown in Table 1.2, which reports scores for a sample of male and female subjects across the two age categories Despite having multiple IVs, the univariate design examines only one dependent variable at a time, ensuring focused analysis of the specific outcome.

The multivariate approach is one in which the research design involves more than one DV For example, apart from investigating whether there is gender

Problem-Solving Scores as a Function of Gender

The study investigates gender differences in problem-solving skills, mathematics skills, English language skills, and overall GPA among students Data shows that males scored 65, 72, 59, and 72 across four assessments, while females scored 76, 65, 84, and 82, highlighting potential variations in performance The research employs a multivariate design, analyzing multiple dependent variables simultaneously to account for their interrelationships This approach enables a comprehensive understanding of how gender may influence various academic skills, with results presented in Table 1.3, which details the scores for problem-solving, mathematics, English language skills, and GPA based on gender.

A major advantage of the multivariate design is that it is often used in

Repeated-measures studies are essential for evaluating the effectiveness of intervention strategies through pre- and post-assessments This study design allows researchers, such as educators, to measure changes in student performance before and after implementing new teaching methods For example, teachers can use repeated-measures approaches to determine whether a new learning strategy introduced by the school leads to significant improvements in students' academic outcomes.

Problem-Solving Scores as a Function of Gender and Age

Problem-Solving Skills, Mathematics Skills, English Language

Skills, and GPA as a Function of Gender

Problem-Solving Skills Maths English GPA

A simple multivariate repeated-measures design can effectively assess changes in GPA over time For instance, Table 1.4 demonstrates that a sample of five students experienced an increase in their post-strategy GPA scores compared to their pre-strategy scores after implementing a new learning strategy This indicates that the innovative approach positively impacted student academic performance Using such research methods helps educators understand the effectiveness of new teaching strategies on student achievement.

Correlational Design

The correlational approach investigates whether a relationship exists between two variables and determines its magnitude and direction Unlike experimental designs that involve manipulating variables, the correlational method examines naturally occurring data without manipulation, such as comparing problem-solving abilities across gender groups For example, a developmental psychologist might study the relationship between age and height in children by recording each child's age and height and calculating the correlation coefficient This coefficient, ranging from -1 to +1, indicates both the strength and direction of the linear relationship between the variables, providing valuable insights into their association.

Hypothesis Testing and Probability Theory

The primary function of hypothesis testing is to see whether our prediction/hypothesis about some social/psychological phenomenon is supported

Pre- and Post-GPA Scores as a Function of a New Learning

In scientific research, conducting experiments and testing hypotheses are fundamental steps Researchers collect, analyze, and interpret data to determine whether the findings support or refute their predictions, which are often derived from existing theories or previous research For example, GPA scores across different semesters show variations such as 2.8 to 3.4 in semester 1 and 3.1 to 3.3 in semester 2, highlighting the importance of analyzing data to assess academic performance trends Ultimately, data-driven analysis enables scholars to validate their hypotheses and advance knowledge in their field.

We use probability to guide decision-making in hypothesis testing, determining whether to accept or reject a hypothesis based on calculated chance probabilities When the probability of obtaining a test result by chance is very low, we are likely to reject the null hypothesis, attributing the result to the manipulation of the independent variable (IV) Conversely, if the probability is high, we tend to retain the null hypothesis, considering the result as a product of chance variation rather than a true effect of the IV This approach helps researchers assess the significance of their findings and make informed conclusions.

Scientific methodology relies on techniques like random sampling, selecting appropriate research designs, and understanding probability theory and hypothesis testing Statistics play a crucial role as a key tool, enabling researchers to draw accurate inferences and conclusions from their data.

Definition of Statistics

Statistics is the science of collecting and analyzing data to draw meaningful conclusions, making it a vital branch of mathematics dedicated to the classification, interpretation, and understanding of numerical facts It enables researchers to interpret large, complex datasets that are impossible to analyze through ordinary observation, providing insights through quantifiable likelihood and probability The field is divided into descriptive statistics, which summarizes data effectively, and inferential statistics, which makes predictions and generalizations based on data analysis.

Descriptive statistics is the branch of statistics that involves organizing, displaying, and understanding data More specifically, descriptive statistics

The scientific methodology of research involves analyzing data to describe, visualize, and summarize information in a meaningful way, enabling the identification of patterns Descriptive statistics plays a crucial role in this process by providing a clear overview of the data without allowing researchers to draw conclusions beyond the analyzed information or test hypotheses It serves as an essential tool for accurately describing data before moving on to more advanced analytical methods.

Descriptive statistics are essential in presenting research findings, as they transform raw data into a more understandable format, especially when dealing with large datasets By summarizing data through measures like averages and distribution, descriptive statistics facilitate easier interpretation and visualization of key patterns For example, calculating the average IQ score of 100 students and analyzing the spread of these scores provides meaningful insights Typically, two main types of descriptive statistics are used to effectively describe data, enhancing clarity and comprehension in research reports.

Measures of central tendency are statistical tools used to identify the central position within a dataset's frequency distribution They provide a summary of the data pattern by highlighting the most typical or representative value Understanding these measures helps analyze the overall trend and distribution of data, making it easier to interpret complex information effectively.

IQ scores scored by the 100 students from the lowest to the highest

The central position of a data set can be effectively described using key statistics such as the mean, median, and mode For instance, the mean represents the average IQ score, illustrating the central tendency of the distribution In the case of 100 IQ scores, the mean provides a clear indication of the overall average, helping to understand the typical IQ score within the dataset.

Measures of spread describe how data scores are dispersed within a dataset, providing insight into variability For example, while the average IQ score for 100 students might be 118, individual scores will vary around this mean Measures such as range, variance, and standard deviation quantify this spread, helping us understand the degree of diversity in the data These statistics are essential for summarizing how tightly or loosely the scores are clustered around the average.

Descriptive statistics focuses on organizing, displaying, and summarizing data, providing a clear overview of what the data shows In contrast, inferential statistics involves drawing meaningful conclusions about a larger population based on sample data, enabling researchers to make predictions and test hypotheses The primary purposes of inferential statistics are to estimate population attitudes or opinions from sample data and to determine whether observed differences between groups, such as gender differences in problem-solving ability, are statistically significant or due to chance While descriptive statistics describes what is present in a data set, inferential statistics extends this to make broader generalizations and informed decisions, which is essential for effective research analysis.

Introduction to SPSS

Learning How to Use the SPSS Software Program

When conducting quantitative research, data collection and analysis are essential for answering research questions and testing hypotheses Introductory social sciences students often rely on textbooks that provide step-by-step instructions for statistical procedures, following a 'cookbook' approach While this method helps students perform calculations, it rarely teaches the meaning behind the mathematical functions or their application in deriving results Consequently, students may learn how to perform statistical tests but lack understanding of the underlying concepts, limiting their overall statistical education Relying solely on procedural steps without grasping the rationale behind them diminishes the depth of students' learning experience in statistics.

The primary goal of statistical analysis is accurate interpretation of results in relation to the research hypotheses Effective interpretation relies on understanding the findings, regardless of whether they are obtained manually or through specialized statistical software Ultimately, what matters most is that the results are correct and properly interpreted, ensuring valid conclusions regardless of the chosen method of analysis.

This book advocates a novel approach for teaching statistical analysis to introductory social sciences students by emphasizing practical use of SPSS, rather than the traditional manual, step-by-step method Learning how to operate SPSS offers distinct advantages, as it helps students focus on interpreting results rather than just following procedural steps The conventional "cookbook" method often results in students blindly following instructions without understanding the rationale behind calculations, which limits their comprehension Instead, mastering SPSS enables students to perform analyses efficiently and concentrate on understanding the meaning of their results, making statistical education more meaningful and applicable.

Learning SPSS software at the introductory level offers a significant advantage, as most social sciences students will use this program throughout their academic careers SPSS is among the most popular statistical packages available, making early familiarity essential By learning to use SPSS early on, students gain valuable experience and confidence in conducting complex analyses later in their studies, enhancing their research skills and academic success.

Introduction to SPSS

When SPSS, Inc., was conceived in 1968, it stood for Statistical Package for the

Since IBM's acquisition of SPSS in 2009, the company has continued to use the SPSS brand to represent its core predictive analytics products IBM defines predictive analytics as tools that transform data into actionable insights by reliably forecasting current conditions and future events This integration emphasizes SPSS's role in helping organizations make data-driven decisions and optimize strategies effectively.

SPSS is a comprehensive software suite specifically designed for analyzing social sciences data It is among the most widely used statistical packages available today, renowned for its user-friendly interface and powerful analytical capabilities The program’s popularity is driven by its ability to efficiently process complex data, generate clear and interpretable results, and support various statistical techniques essential for social science research.

• Allows for a great deal of flexibility in the format of data

• Provides the user with a comprehensive set of procedures for data transformation and file manipulation

• Offers the researcher a large number of statistical analyses commonly used in the social sciences

SPSS is an essential and powerful statistical software for both beginners and advanced researchers, offering relative ease of use after learning basic concepts The Windows version features a user-friendly point-and-click interface, enabling users to perform analyses by simply navigating through dialog boxes, which eliminates the need to learn complex syntax This intuitive interface has made SPSS highly popular among researchers with little interest in coding However, SPSS for Windows also retains the syntax method, allowing advanced users to execute analyses by writing and executing command scripts for greater flexibility and control.

Beginners often ask which SPSS method is more effective, comparing the Windows interface and syntax-based approaches Both methods have their unique advantages and disadvantages, which will be explored in Section 2.3 to help researchers choose the most suitable option for their analysis.

This chapter is designed for beginner students and researchers, offering a clear overview of SPSS's two fundamental functions It explains how to set up data files in SPSS for Windows and guides users through conducting analyses using both the Windows interface and syntax commands, ensuring a comprehensive understanding of essential SPSS operations.

A researcher conducted a survey to assess the level of support among Thai people for four types of euthanasia: active euthanasia, passive euthanasia, voluntary euthanasia, and non-voluntary euthanasia The survey responses, collected through a structured questionnaire as shown in Table 2.1, provide insights into public opinions regarding these different forms of mercy killing.

Before data entry, it is essential to prepare a comprehensive codebook that includes variable names from the questionnaire, their corresponding SPSS variable names, and detailed coding instructions This codebook serves as a vital tool for researchers to efficiently track and manage all survey variables and ensure consistency between the questionnaire and the SPSS data file For example, Table 2.2 provides the complete codebook for the questionnaire outlined in Table 2.1, facilitating accurate and organized data analysis.

Table 2.3 presents the responses (raw data) obtained from a sample of 10 respondents to the euthanasia questionnaire.

The following steps demonstrate how the data presented in Table 2.3 are entered into a SPSS data file.

1 When the SPSS program (Version 23) is launched, the following window will open.

Since the purpose of the present exercise is to create a new data file, close this window by clicking The following Untitled1

[DataSet0] – IBM SPSS Statistics Data Editor screen will then be displayed.

2 Prior to data entry, the variables in the data set must be named and defined In the Untitled1 [DataSet0] – IBM SPSS Statistics Data

Editor screen, the names of the variables are listed down the side

(under the Name column), with their characteristics listed along the top (Type, Width, Decimals, Label, Values, Missing, Columns,

The codebook outlined in Table 2.2 serves as a comprehensive guide for naming and defining variables, ensuring clarity and consistency in data analysis For instance, the first variable, GENDER, is clearly coded with 1 representing male, facilitating accurate data categorization and interpretation This standardized coding system enhances the reliability of the measurement process, supporting effective alignment of data collection and analysis efforts.

2 = female Thus, in the first cell under Name in the Data Editor screen, type in the name GENDER To assign the coded values

(1 = male, 2 = female) to this variable, click the corresponding cell under Values in the Data Editor screen Click the shaded area to open the following Value Labels window.

The euthanasia survey questionnaire collects demographic data, including gender (male or female) and age in years Participants are asked to evaluate their support for four euthanasia types—active euthanasia, passive euthanasia, voluntary euthanasia, and non-voluntary euthanasia—by carefully considering each of the four statements related to these issues Respondents indicate their level of support by circling a number on a 6-point scale for each statement, providing insights into public attitudes towards different forms of euthanasia.

Strongly Moderately Barely Barely Moderately Strongly not support not support not support support support support

Age Age Age in years

Active euthanasia Active 1 = strongly not support

Passive euthanasia Passive 1 = strongly not support

Voluntary euthanasia Voluntary 1 = strongly not support

6 = strongly support Non-voluntary euthanasia Non-Voluntary 1 = strongly not support

3 In order to define the code for male respondents, type 1 in the Value: cell, and in the Label: cell, type Male Next, click to complete the coding for the male respondents For female respondents, type

To code female respondents, select "Female" in the Label: cell and input the appropriate value in the Value: cell After entering the data, click to finalize the coding process Once complete, the Value Labels window will display the updated labels, confirming that the female respondents have been correctly coded This step ensures accurate categorization for data analysis and maintains consistency within your dataset.

Next, click to complete the coding for the GENDER variable and to return to the Untitled1 [DataSet0] – IBM SPSS Statistics

4 Repeat the above coding procedure for the rest of the variables in the codebook Please note that the AGE variable is a continuous variable and therefore has no coded values.

5 If the researcher wishes to attach a label to a variable name (to provide a longer description for that variable), this can be done by typing a label in the corresponding cell in the Label column For example, the researcher may wish to attach the label ‘support for active euthanasia’ to the variable ACTIVE This label will be printed in the analysis output generated by SPSS The following Untitled1

[DataSet0] – IBM SPSS Statistics Data Editor screen displays the names of all the variables listed in the code book, and where relevant, their Labels and Values codes.

Data can only be entered via the Data View screen Switch the present Variable

To access the Data View, click on the Data View tab located at the bottom left corner of the screen In the Data View screen, each row represents a respondent, while each column corresponds to a specific variable Start by entering the data from Table 2.3 into the first data cell (row 1, column 1), then continue inputting data accordingly The Data View displayed after data entry shows the responses collected from all 10 respondents.

2.2.6 Saving and Editing Data File

Once data entry is completed, the data file can be saved From the menu bar, click File, then Save As Once it has been decided where the data file is to be saved to, type a name for the file As this is a data file, SPSS will automatically append the suffix.SAV to the data file name (e.g., TRIAL.SAV).

To edit an existing file, click File, then Open, and then Data from the menu bar Scroll through the names of the data files and double-click on the data file to open it.

SPSS Analysis: Windows Method versus Syntax Method

Once the SPSS data file has been created, the researcher can conduct the chosen analysis either through the Windows method (point-and-click) or the syntax method The primary advantage of using the Windows method is clearly its ease of use With this method, the researcher accesses the pull-down menu by clicking in either the Data View or Variable View mode, and then point-and-clicks through a series of windows and dialogue boxes to specify the kind of analysis required and the variables involved There is no need to type in any syntax or commands to execute the analysis Although this procedure seems ideal at first, paradoxically it is not always the method of choice for the more advanced and sophisticated users of the program Rather, there is clearly a preference for the syntax method among these users This preference stems from several good reasons from learning to use the syntax method.First, when conducting complex analysis, the ability to write and edit syntax commands is advantageous For example, if a researcher mis-specifies a syntax command for a complex analysis and wants to go back and rerun it with minor changes, or if the researcher wishes to repeat an analysis multiple times with minor variations, it is often more efficient to write and edit the syntax command directly than to repeat the Windows pull-down menu sequences Second, from my teaching experience with SPSS, I believe that students have a better “feel” for statistics if they have to write syntax commands to generate the specific statistics they need, rather than merely relying on pull-down menus In other words, it provides a better learning experience Finally, and perhaps most important, several SPSS procedures are available only via the syntax method.

SPSS Analysis: Windows Method

After entering the data, the researcher can proceed with the data analysis process If the goal is to obtain general descriptive statistics for all variables in the dataset TRIAL.SAV, the analysis will provide essential insights into the data's overall characteristics Descriptive statistics, such as means, standard deviations, and frequency distributions, help summarize the data effectively Conducting this step ensures a comprehensive understanding of the variables before further statistical testing or modeling Utilizing these descriptive measures in statistical analysis enhances data interpretation and informs subsequent research decisions.

1 From the menu bar, click Analyze, then Descriptive Statistics, and then Frequencies The following Frequencies window will open.

2 In the left-hand field containing the study’s six variables, click (highlight) these variables, and then click to transfer the selected variables to the Variable(s): field.

3 Click to open the Frequencies: Statistics window below Suppose the researcher is only interested in obtaining statistics for the Mean, Median, Mode, and Standard Deviation for the six variables In the Frequencies: Statistics window, check the fields related to these statistics Next click

4 When the Frequencies window opens, run the analysis by clicking

See Table 2.4 for the results.

SPSS Analysis: Syntax Method

/STATISTICS=MEAN MEDIAN MODE STDDEV.

1 From the menu bar, click File, then New, and then Syntax The following IBM SPSS Statistics Syntax Editor window will open.

2 Type the Frequencies analysis syntax command in the IBM SPSS

Statistics Syntax Editor window If the researcher is interested in obtaining all descriptive statistics (and not just the mean, median, mode, and standard deviation), then replace the syntax:

/STATISTCS=MEAN MEDIAN MODE STDDEV. with

3 To run the Frequencies analysis, click or click and then All.

(NOT E : Table E presents a glossary of the SPSS syntax files employed for all the examples in this book).

Support for Non- Voluntary Euthanasia

Std Deviation 0.52705 6.66667 0.63246 0.67495 0.69921 0.51640 a Multiple modes exist The smallest value is shown.

Frequency Table Gender Frequency Percent Valid Percent Cumulative Percent

Age Frequency Percent Valid Percent Cumulative Percent

The Statistics Table presents the requested Mean, Median, Mode, and

Standard deviation statistics are calculated for five continuous variables: AGE, ACTIVE, PASSIVE, VOLUNTARY, and NON-VOLUNTARY Since these variables are measured at least at the ordinal level, their mean, median, and standard deviation provide meaningful insights In contrast, the Gender variable is categorical, making its mean, median, and standard deviation statistics not meaningful or interpretable for analysis.

The survey results indicate that the 10 respondents have a mean age of 26 years and a median age of 25 years The data shows a diverse age distribution with no single age occurring more frequently than others, as reflected in the SPSS output.

Support for Active Euthanasia Frequency Percent Valid Percent Cumulative Percent

Support for Passive Euthanasia Frequency Percent Valid Percent Cumulative Percent

Support for Voluntary Euthanasia Frequency Percent Valid Percent Cumulative Percent

Support for Non-Voluntary Euthanasia Frequency Percent Valid Percent Cumulative Percent

In our analysis using SPSS, the most common (mode) lowest age value recorded was 18 years The standard deviations for the four euthanasia variables indicate the variability in responses: ACTIVE euthanasia has a standard deviation of 0.63, PASSIVE euthanasia 0.67, and VOLUNTARY euthanasia 0.70, reflecting the degree of spread around the mean scores for each category.

NON-VOLUNTARY: SD = 0.52 For the AGE variable, the standard deviation shows that its average deviation from the mean is 6.67 years.

Support for euthanasia varies significantly across different types, with voluntary euthanasia receiving the highest support (Mean = 5.40; Median = 5.50), indicating strong public acceptance Passive euthanasia follows, with a mean of 3.70 and a median of 4.00, reflecting moderate support In contrast, active euthanasia has the lowest support (Mean = 1.80; Median = 2.00), and non-voluntary euthanasia receives the least support (Mean = 1.40; Median = 1.00) The frequency table details the distribution of responses across six variables—GENDER, AGE, ACTIVE, PASSIVE, VOLUNTARY, and NON-VOLUNTARY euthanasia—highlighting variations in attitudes within different demographic groups.

The table displays the frequency of each variable’s values, their percentage of the total sample, valid percentages excluding missing cases, and cumulative percentages for each value For example, the gender variable shows an equal distribution with 5 males and 5 females, each comprising 50% of the sample, and with no missing data, the percentage and valid percentage are identical Cumulative percentages accumulate the proportion of respondents at or below each value, so in this case, 50% of respondents are male, and combined with females, the total reaches 100%.

The frequency Tables for the variables of AGE, ACTIVE, PASSIVE,

Voluntary and non-voluntary euthanasia are interpreted identically For instance, when considering support for active euthanasia, 30% of respondents (3 individuals) strongly opposed it, while an additional 60% (6 individuals) also opposed, bringing the total opposing support to 90%.

‘moderately not support’, and 1 respondent (10%; 100 cumulative percent) responded with ‘barely not support’.

Descriptive Statistics 3 Basic Mathematical Concepts and Measurement

Basic Mathematical Concepts

Many students in social sciences experience unnecessary anxiety about learning statistics, often due to the misconception that they must master complex mathematical formulas In reality, mastering basic statistical principles does not require being a mathematical genius or expert in calculus While some mathematics, such as elementary algebra and basic operations like addition, subtraction, multiplication, division, squares, and square roots, is essential for success in statistics courses Most of these mathematical skills are typically learned during high school, making statistics accessible to students with foundational math knowledge Overcoming the fear of mathematics is key to building confidence and succeeding in social science statistics courses.

Understanding the key symbols used in statistics is essential for students to grasp the subject more effectively Mastering common symbols such as X, Y, N, and Σ significantly eases the learning process, as these appear frequently throughout statistical studies Although students will encounter many symbols in their statistics journey, focusing on these four core symbols will provide a solid foundation for understanding statistical concepts and procedures.

In research studies, the symbols X and Y are commonly used to denote measured variables, with X typically representing one variable and Y another For example, if a study measures IQ and age, X might be used to symbolize IQ, while Y represents age Table 3.1 illustrates the distribution of these two variables across five subjects, providing valuable data for analysis.

In this example, each of the five X scores corresponds to specific IQ values, while each of the five Y scores represents specific age measurements These scores are distinguished by their subscripts, indicating the particular subject associated with each value This detailed scoring system allows for precise analysis of individual IQ and age data, facilitating accurate interpretation of the results.

In this study, X₁ represents the IQ score of the first subject, which is 118, while X₂ denotes the IQ score of the second subject, recorded as 105 Similarly, Y₁ indicates the age score of the first subject, aged 22, and Y₂ corresponds to the age score of the second subject, aged 19 These variables are used to analyze the relationship between IQ scores and age among the subjects, highlighting key data points for understanding cognitive performance across different age groups.

• Another frequently used mathematical notation is the symbol N, which represents the number of subjects in the data set Thus, for the example above, there are 5 subjects,

• A symbol that is used most frequently to denote the summation of all or part of the scores in the data set is Σ(sigma) The full algebraic summation phrase is i

In summation notation, the expressions above and below the summation sign specify which scores to include The notation below the summation sign indicates the starting score, while the notation above indicates the ending score For example, this algebraic phrase instructs us to sum the scores from the first score (X₁) to the Nth score (X₅) Applying this to IQ data, we sum the relevant scores within the specified range to analyze the overall performance.

Distribution of IQ (X) and Age (Y) Scores

Basic Mathematical Concepts and Measurement

When the summation is across all the scores in the distribution, then the full algebraic summation phrase can be summarized as

In data analysis, it is not always necessary to sum across the entire distribution of scores from 1 to N Instead, specific subsets of scores, such as the second, third, and fourth values, can be selectively summed to focus on particular data points This targeted summation approach allows for more precise insights and can be efficiently expressed using algebraic summation notation, such as summing only the desired indices like i = 2, 3, and 4 Understanding how to perform selective summation is essential for accurate data interpretation and effective statistical analysis.

Measurement Scales (Levels of Measurement)

When conducting a statistical analysis, it is essential for researchers to consider how variables are measured, as different measurements can be on various scales such as nominal, ordinal, interval, or ratio Understanding the measurement scale of each variable helps interpret the data accurately and determines which statistical tests are appropriate Variables in behavioral sciences are typically classified into these four scales, which differ based on attributes like magnitude, equal intervals between units, and the presence of an absolute zero point Properly identifying the measurement scale is crucial for meaningful data analysis and valid conclusions.

Nominal scales represent the lowest level of measurement, classifying variables into distinct categories without implying any order or magnitude Examples include gender, hair color, religion, and ethnicity, where individuals are grouped into categories such as American, Chinese, Australian, African, or Indian Once categorized, all individuals within a particular group are assumed to be equivalent on the measured attribute, even if they differ in specific characteristics Assigning numbers to these categories (e.g., codes for ethnicity) facilitates data organization but does not indicate any quantitative relationship among categories.

1 = American, 2 = Chinese, 3 = Australian, 4 = African, and 5 = Indian), but the numbers are only used to name/label the categories They have no magnitude in terms of quantitative value.

Ordinal measurement involves ranking or ordering the variable being studied, such as customer satisfaction levels on a 4-point scale from ‘1 = very dissatisfied’ to ‘4 = very satisfied,’ indicating an ordered ranking from least to most satisfied While ordinal scales allow differentiation between rankings, they do not specify the magnitude of differences between ranks, meaning the intervals between categories are not meaningful For example, a customer rating ‘4’ is more satisfied than one rating ‘3,’ but the scale does not reveal how much more satisfied they are This measurement method enables researchers to quantify relative positions but cannot assess exact differences in satisfaction levels Proper understanding of ordinal data is essential for accurate analysis and interpretation in market research and customer feedback studies.

Ordinal scales do not capture the true magnitude of differences between rankings, as the gap between ‘very dissatisfied’ and ‘dissatisfied’ likely differs from that between ‘dissatisfied’ and ‘satisfied.’ This means that the differences on an ordinal scale cannot be assumed to reflect equal psychological distances, limiting their effectiveness in accurately measuring concepts like customer satisfaction.

Interval measurement allows for the precise specification of the distance between two stimuli along a given dimension, with scale intervals maintaining consistent interpretation throughout Unlike ordinal scales, where the differences between rankings such as 'very satisfied' (rank 4) and 'satisfied' (rank 3) do not necessarily represent equal distances, interval scales provide meaningful and equal intervals, enabling accurate comparisons and analyses.

In a satisfaction survey, scores ranked 1 (very dissatisfied) and 2 (dissatisfied) indicate low levels of contentment On an interval scale, equal differences in scale values represent equal differences in meaning, such as in exam scores where a 40-point difference signifies the same level of change regardless of position on the scale For instance, the difference between scores of 80 and 40 is equivalent to the difference between 60 and 20, highlighting the uniformity in measurement However, it is incorrect to interpret these scores as representing proportional skills—for example, assuming a score of 80 is twice as good as a score of 40—because interval scales lack a true zero point indicating the absence of the measured attribute A student scoring zero does not necessarily lack all statistical skills; the score could reflect the exam’s difficulty or other external factors rather than complete skill absence.

The ratio scale of measurement replaces the arbitrary zero point of the interval scale with a true zero that indicates the absence of the measured variable, providing more meaningful data It is the most informative scale because it combines the features of nominal, ordinal, and interval scales The ratio scale functions as a nominal scale by assigning labels or categories to objects, as an ordinal scale by ranking these objects, and as an interval scale by ensuring that equal differences between values hold the same meaning at different points on the scale.

A ratio scale is characterized by having a true zero point and equal intervals between measurement units, allowing meaningful comparisons of ratios For example, weight is a ratio scale because zero weight indicates an absence of weight, and the intervals between measurements are equal This means that the difference between 10 g and 15 g is the same as between 45 g and 50 g, and an object weighing 80 g is twice as heavy as one weighing 40 g Using ratio scales enables us to express that one variable is twice, half, or multiple times as much as another, providing precise and meaningful quantitative comparisons.

Types of Variables

In scientific experiments, variables are classified as independent variables (IVs) and dependent variables (DVs) The independent variable is the factor that researchers manipulate or control to assess its impact, while the dependent variable is the outcome affected by changes in the IV Understanding the relationship between IVs and DVs is essential for accurately analyzing experimental results, as the DV depends on the manipulation of the IV.

A developmental psychologist might investigate whether gender influences problem-solving skills by comparing two groups—males and females—where gender serves as the independent variable (IV) The problem-solving scores of these groups are recorded as the dependent variable (DV), allowing researchers to analyze potential differences based on gender This approach helps in understanding the impact of gender on cognitive abilities, contributing to insights in developmental psychology.

Variables in research can be categorized as either independent (IV) or dependent (DV), and further classified as continuous or discrete Continuous variables, like weight, can take on an infinite number of possible values between measurement units—for example, between 1 and 2 grams, values can range from 1.1 to 1.689 grams or any value in between Understanding the distinction between continuous and discrete variables is essential for accurate data collection and analysis in scientific studies.

Discrete variables are those that can only take on specific, fixed values with no possible intermediate values; for example, the number of cars on a road can be 1, 2, 3, or 4 cars, but not 1.26 or 2.56 cars This characteristic means the value of a discrete variable changes in fixed increments, with no possible values between the adjacent units.

3.3.3 Real Limits of Continuous Variables

A continuous variable can take on an infinite number of possible values between adjacent points on a scale, making precise measurement inherently challenging When measuring something like time, it is important to recognize that such measurements are only approximations due to the limitations of measuring instruments Enhanced sensitivity in measurement tools can slightly improve accuracy, but exact values remain elusive Typically, the true limits of a measurement are considered to fall within the recorded number plus or minus (±) a margin of error.

When measuring weight, such as stepping on a calibrated bathroom scale that displays readings in 1-kilogram units, the indicated value (e.g., 56.6 kg) represents an approximation due to the continuous nature of weight Although this measurement provides a specific value, it is more accurate to specify the interval within which the true weight likely falls, acknowledging the inherent measurement limitations of continuous variables.

Understanding basic mathematical concepts and measurement is essential for accurate weight assessment Measurement is defined as the value plus or minus (±) half the unit, meaning your true weight should fall within a specific range For instance, if your weight is measured at 56.6 kg with a measurement unit of 1 kg, your true weight must be between 56.1 kg and 57.1 kg This approach ensures precise estimates by accounting for measurement variability.

56.6 ± 0.5) These numbers are called the real limits of that measure.

When reporting numerical results in research, it is essential to determine the appropriate number of decimal places and the value of the last digit to ensure accuracy and clarity The process involves answering two straightforward questions: how many decimal places should the final answer include, and what value should the final digit have? Adhering to simple rules for these questions helps maintain consistency and precision in scientific reporting, ultimately enhancing the credibility of the results.

The decision on how many decimal places to report in the final answer depends on the researcher's discretion, with options typically including two, three, or four decimal places.

Second, in answering the question of what value should the last digit have, follow these steps:

• Decide which is the last digit to keep

• Leave it the same if the next digit is less than 5 (this is called rounding down)

• Increase it by 1 if the next digit is 5 or more (this is called rounding up)

Table 3.2 shows how numbers are rounded up or down to two decimal places.

Rounding Numbers Up or Down to Two Decimal Places

Frequency Distributions

Frequency distributions are visual displays that organize and present frequency counts so that the information can be interpreted more easily

Frequency distributions display how often specific quantities or groups of quantities appear within a dataset, providing valuable insights into data patterns For instance, the age frequency distribution reveals the number of individuals at each age level, such as 26 years old The frequency (f) represents the count of occurrences for a particular age, while the distribution (d) illustrates the pattern of these frequencies across different ages Analyzing age distributions helps identify common age groups within a population, making it a vital tool for demographic and statistical analysis.

Frequency distributions are essential tools that help researchers interpret collected data effectively For example, if you are one of 100 individuals who took an IQ test and scored 120, this number alone provides little insight without a standard for comparison Understanding your IQ score's significance requires comparing it to others—such as knowing how many individuals scored higher or lower than you Table 4.1 displays the raw IQ scores from all participants, illustrating the distribution and enabling meaningful analysis of the data.

Presenting IQ scores randomly, even when all 100 scores are displayed, makes it difficult to interpret the data effectively A more informative approach is to use a frequency distribution, which organizes the IQ scores based on how often each score occurs This method provides clearer insights into the data, highlighting patterns and the distribution of intelligence levels across the sample Utilizing frequency distribution charts enhances data comprehension and offers valuable implications for understanding overall cognitive performance.

To conduct a frequency analysis on the 100 IQ scores, choose either the

SPSS Windows method or the SPSS Syntax File method.

The data set has been saved under the name: EX1.SAV

1 When the SPSS program (Version 23) is launched, click Analyze on the menu bar, then Descriptive Statistics, and then Frequencies The following Frequencies window will open.

2 In the left-hand field containing the study’s IQ variable, click (highlight) this variable, and then click to transfer the selected IQ variable to the Variable(s): field.

3 As the purpose of this analysis is to obtain a frequencies distribution for the 100 IQ scores (i.e., you are not interested in obtaining statistics for the variable’s mean, median, mode, and standard deviation), click to run the analysis See Table 4.2 for the results.

1 From the menu bar, click File, then New, and then Syntax The following IBM SPSS Statistics Syntax Editor Window will open.

IQ Frequency Percent Valid Percent Cumulative Percent

Table 4.2 displays the ranking of IQ scores from the lowest to the highest It includes detailed information on each score’s frequency count, percentage, valid percentage, and cumulative percentage, providing a comprehensive overview of the IQ distribution.

• The frequency count is simply the number of times that score has occurred.

The percentage indicates how often a specific score occurs relative to the total number of scores, including any missing data For example, if there are 4 missing IQ scores out of 100 total scores, the percentage for a score of 109 with a frequency of 6 is calculated based on the entire total of 100 cases This means that the 6% reflects the proportion of that score's occurrence out of all 100 cases, regardless of missing data.

The valid percent represents the frequency of a specific score expressed as a percentage of the total valid scores, excluding any missing data For example, if there are 96 valid cases after accounting for 4 missing IQ scores, the valid percentage for an IQ score of 109, which occurs 6 times, is calculated based on these 96 valid cases This metric helps provide an accurate understanding of the distribution of valid scores within the dataset.

IQ Frequency Percent Valid Percent Cumulative Percent

Frequency Distributions of the total scores’ frequency, exclusive of the 4 missing cases (i.e.,

The cumulative percent helps determine the proportion of scores that fall above or below a specific value For instance, in Table 4.2, an IQ score of 91 has a cumulative percent of 10%, indicating that all scores from 81 to 91 inclusive account for 10% of the total IQ scores This measure sums all valid percent values up to and including the target score, providing researchers with a clear understanding of score distribution within the dataset.

A frequency distribution helps organize data to improve understanding and interpretation For example, with a score of 120, 33 students (33%) have higher IQ scores, while 64 students (64%) scored lower, making it easy to analyze relative performance Additionally, only 3 students have an IQ score of exactly 120, illustrating how this approach simplifies identifying the distribution of scores within a dataset.

Table 4.2 presents the ungrouped frequency distribution of 100 IQ scores, illustrating how listing individual scores can lead to many with low frequencies and a challenging visualization of the data's central tendency When scores are widely spread out, grouping them into intervals simplifies analysis and interpretation Creating class intervals of equal width and calculating the frequency of scores within each interval results in a grouped frequency distribution, making it easier to identify central tendencies and overall data patterns.

4.2.1 Grouping Scores into Class Intervals

Grouping scores into class intervals involves consolidating individual data points into defined ranges, which simplifies the data distribution but results in a loss of detailed information The wider the class intervals, the greater the information loss, potentially reducing the precision of analysis For example, IQ scores can be grouped into broader categories, such as in Table 4.3, where 100 individual scores are summarized within specific class intervals to facilitate easier interpretation.

5 classes with an interval width of 20 units wide.

Although grouping data into wide intervals has prevented low frequency counts of 1, it has led to significant information loss Specifically, individual scores lose their identities when aggregated into class intervals, resulting in unavoidable errors in statistical estimation based on grouped scores For example, there are 51 scores within the 101–120 interval, but the distribution of these scores within this range remains unclear.

Do they all fall at 101? Or at 120? Or are they distributed evenly across the interval? The answer is that we don’t know as that information has been lost

Choosing the appropriate interval width is crucial in frequency distribution analysis The intervals should not be so broad that they obscure important details, nor so narrow that they undermine the benefits of grouping data Typically, researchers aim for 10–20 class intervals to balance detail and clarity, with the exact number depending on the range and number of raw data scores, such as IQ scores This approach ensures effective data representation in behavioral sciences.

4.2.2 Computing a Frequency Distribution of Grouped Scores

Let’s suppose that we have decided to group the data (the 100 IQ scores) into approximately 10 class intervals The procedural steps to be employed are as follows:

Step 1 Find the range of the scores The range is simply the difference between the highest and lowest score values contained in the original data That is,

To analyze the distribution of scores, first calculate the range by subtracting the lowest score (74) from the highest score (81), resulting in a range of 7 Next, determine the interval width by dividing the range (7) by the desired number of class intervals (approximately 10), which helps create a clear and organized frequency distribution This method ensures each class interval effectively represents a segment of the data, facilitating better data analysis and interpretation.

. i= Range = Number of class intervals

IQ Scores from Table 4.2 Grouped into Class Intervals of 20 Units Wide

When the resulting value is not a whole number, applying the rounding rule is essential Since the decimal part (0.4) is less than 0.5, the value is rounded down to the nearest whole number, which is 7 Therefore, in this example, the value of i is determined to be 7.

Graphing

Frequency distributions are commonly displayed in graphical formats because visual representations significantly enhance data comprehension Using charts such as bar graphs, histograms, frequency polygons, and cumulative percentage curves simplifies complex survey data, making it easier to interpret These visual tools effectively condense large data sets into clear, intuitive formats that highlight key insights, improving overall understanding of research results.

Bar graphs are commonly used to visualize frequency distributions of nominal or ordinal data, with each bar representing a specific category such as gender (male, female) or ethnicity (Chinese, Thai, Australian) The height of each bar reflects the quantity or frequency within that category, making bar graphs ideal for comparing values across different groups It is important to note that for categorical data, there is no quantitative relationship between categories, which is visually emphasized by the gaps between bars in the graph This article also explains how to construct a bar graph using both the SPSS Windows interface and SPSS Syntax methods for accurate data visualization.

Suppose that in a survey study, the following question was asked:

What is your current employment status?

In this example, respondents are asked to select the employment category that best describes their employment status, which helps capture relevant employment data These six employment categories are consolidated under the variable EMPLOY, enabling efficient analysis of employment patterns across different populations Using clear and specific categories ensures accurate data collection and important insights into labor market trends Proper categorization under EMPLOY enhances the quality of employment statistics for research and policy development.

1 In order to draw a bar graph of the frequency distribution representing the EMPLOY variable, open the data set EX6.SAV Click Graphs on the menu bar, then Legacy Dialogs, and then Bar The following

Bar Charts Window will open.

2 Click (highlight) the icon Make sure that the Summaries for

Groups of Cases bullet-point is checked Click to open the

Define Simple Bar: Summaries for Groups of Cases Window below.

3 When the Define Simple Bar: Summaries for Groups of Cases

Window opens, check the bullet-point N of cases Next, transfer the EMPLOY variable to the Category axis: cell by clicking the EMPLOY variable (highlight) and then

Click to draw a bar graph of the frequency distribution representing the EMPLOY variable (see Figure 5.1).

/BAR(SIMPLE)=COUNT BY EMPLOY.

2 Type the Bar Graph syntax command in the IBM SPSS Statistics

3 To run the Bar Graph analysis, click or click and then All.

Figure 5.1 displays a bar graph illustrating the frequency distribution of the EMPLOY variable To add frequency labels to each bar, double-click on the figure in SPSS to open the Chart Editor Window This allows for easy customization and precise labeling of the chart for clearer data presentation and enhanced readability.

0 Full-time employment Part-time employment Household duties Employment status

Bar graph of the frequency distribution representing the EMPLOY variable.

To display the frequency counts on your bar graph, click on the Show Data Labels icon This action labels each bar in the graph with its corresponding frequency, enhancing data clarity Once you close the data labels window, the bar graph will automatically update to include these counts for each category of the EMPLOY variable, providing a clear visual representation of the distribution (see Figure 5.2).

From Figure 5.2, it can be seen that the majority of the respondents are in full-time employment (N = 10; 50%).

Histograms are commonly used to visualize frequency distributions of interval or ratio data, with the height of each column representing the frequency count within a specific data range Unlike bar charts, which display categorical data such as gender, histograms display class intervals based on quantitative variables like age, weight, or height The key difference is that bar charts use separate columns for categories, while histograms use continuous intervals to depict the distribution of numerical data Both visualizations utilize columns on a graph to effectively illustrate data distribution and frequency patterns.

0 Full-time employment Part-time employment Household duties Employment status

Bar graph and the frequency counts of the distribution representing the EMPLOY variable.

Suppose that you want to draw a histogram of the grouped frequency distribution of the 100 IQ scores presented in Table 4.5.

1 Open the data set EX7.SAV Click Graphs on the menu bar, then

Legacy Dialogs, and then Histogram The following Histogram

2 Transfer the GROUP variable to the Variable: cell by clicking the

GROUP variable (highlight) and then

Click to draw a histogram of the frequency distribution representing the GROUP variable (see Figure 5.3).

Histogram of the frequency distribution representing the GROUP variable.

2 Type the Histogram syntax command in the IBM SPSS Statistics

3 To run the Histogram analysis, click or click and then All.

Figure 5.3 displays the histogram of the frequency distribution for the GROUP variable To add data labels showing each bar's frequency count, double-click on the histogram in SPSS to open the Chart Editor Window This allows for easy customization and clear visualization of the data distribution.

To display the frequency counts on each bar of your histogram, click on the Show Data Labels icon This will label every bar with its corresponding frequency, making your data clearer After closing the window, the histogram will update to show the frequency distribution for the GROUP variable, with each bar labeled accordingly (see Figure 5.4) This step enhances data visualization and helps interpret the frequency data effectively.

From Figure 5.4, it can be seen that the majority of the IQ scores (N = 63) are between the class intervals of 102–108 (GROUP = 4) and 123–129 (GROUP = 7).

Frequency polygons are an effective way to represent frequency distributions of interval or ratio data, using points plotted over the midpoints of class intervals at heights corresponding to their frequencies Unlike histograms, where bars visually display data, frequency polygons connect these points with lines, providing a clear comparison of data trends The horizontal axis of a frequency polygon represents the class intervals of a quantitative variable such as age, weight, or height, making it a useful tool for analyzing the distribution of continuous data.

Suppose you want to draw a frequency polygon of the grouped frequency distribution of the 100 IQ scores presented in Table 4.5.

Histogram and the frequency counts of the distribution representing the GROUP variable.

1 Open the data set EX7.SAV Click Graphs on the menu bar, then

Legacy Dialogs, and then Line The following Line Charts Window will open.

2 Click (highlight) the icon Make sure that the Summaries for groups of cases bullet-point is checked Click to open the

3 When the Define Simple Line: Summaries for Groups of Cases Window opens, check the bullet-point N of cases Next, transfer the

GROUP variable to the Category axis: cell by clicking the GROUP variable (highlight) and then

Click to draw a frequency polygon of the frequency distribution representing the GROUP variable (see Figure 5.5).

/LINE(SIMPLE)=COUNT BY GROUP.

2 Type the Frequency polygon syntax command in the IBM SPSS

3 To run the Frequency polygon analysis, click or click and then All.

The frequency polygon of the GROUP variable is displayed in Figure 5.5 To label each point with its corresponding frequency count, simply double-click on the figure within SPSS This action opens the Chart Editor Window, allowing for detailed customization and annotation of the frequency distribution.

Frequency polygon of the frequency distribution representing the GROUP variable.

To display the frequency counts on your chart, click on the Show Data Labels icon This will label each point on the frequency polygon with its corresponding frequency count, providing clear visibility of data distribution After closing the window, the frequency polygon for the GROUP variable will clearly include the frequency counts for each point, as shown in Figure 5.6.

The frequency polygon results (Figure 5.6) closely resemble the histogram (Figure 5.4), showing that most IQ scores (N = 63) fall within the class intervals of 102–108 (GROUP 4) and 123–129 (GROUP 7).

The cumulative percentage curve, also known as an ogive, represents frequency distribution by calculating the percentage of the cumulative frequency within each interval Cumulative frequency and cumulative percentage graphs are identical in appearance The key advantage of using cumulative percentage over cumulative frequency is that it simplifies the comparison of different data sets, as its y-axis is scaled in percentages from 0% to 100%, enabling easy visual comparison Cumulative percentage is calculated by dividing the cumulative frequency by the total number of observations (n) and multiplying the result by 100, ensuring the final value always reaches 100%.

Suppose you want to draw a cumulative percentage curve of the frequency distribution of the 100 IQ scores presented in Table 4.2.

Group Group Class interval Frequency ( f )

Frequency polygon and the frequency counts of the distribution representing the GROUP variable.

1 Open the data set EX7.SAV Click Graphs on the menu bar, then Legacy

Dialogs, and then Line The following Line Charts Window will open.

2 Click (highlight) the icon Make sure that the Summaries for

Groups of Cases bullet-point is checked Click to open the

3 When the Define Simple Line: Summaries for Groups of Cases Window opens, check the bullet-point Cum % Next, transfer the IQ variable to the Category axis: cell by clicking the IQ variable (highlight) and then

Click to draw a cumulative percentage curve of the frequency distribution representing the IQ variable (see Figure 5.7).

/LINE(SIMPLE)=CUPCT BY IQ.

2 Type the cumulative percentage syntax command in the IBM SPSS

3 To run the cumulative percentage analysis, click or click and then All.

Measures of Central Tendency

6.1 Why Is Central Tendency Important?

Central tendency is essential in social sciences as it provides a single, representative value of a data set, facilitating understanding and analysis While frequency distributions, like the one used for 100 IQ scores, organize data into rank-ordered scores with corresponding frequency counts, they do not offer a concise quantitative summary of the entire distribution Although frequency distributions highlight data variability, they do not enable us to make comprehensive quantitative comparisons or characterize the distribution as a whole Therefore, measures of central tendency are crucial for summarizing data effectively and making meaningful comparisons across different datasets.

A psychologist may explore potential gender differences in IQ scores by analyzing data collected from a sample of 10 males and 10 females The study involves examining the distribution of IQ scores within this sample to identify any notable variations between genders Table 6.1 presents the detailed distribution of IQ scores for these respondents, providing valuable insights into possible gender-based disparities in intelligence test results.

Analyzing the frequency distribution of IQ scores for male and female respondents reveals the difficulty in making definitive quantitative comparisons due to overlapping score ranges To effectively compare the two groups, psychologists typically calculate the average IQ for each group, with this example showing a mean IQ of 103.5 for males and 102.3 for females This comparison demonstrates that, on average, males scored slightly higher than females in this sample Ultimately, using measures of central tendency like average IQ simplifies the process of drawing meaningful conclusions about group differences.

A measure of central tendency is a summary measure that describes an entire set of data with a single value that represents the middle or center of its distribution.

The three primary measures of central tendency are the mean, median, and mode, each serving as valid indicators of a dataset's central point Choosing the most appropriate measure depends on specific data conditions, making it essential to understand their proper application In the upcoming sections, we will explore how to calculate these measures and identify when each one is most suitable for accurate data analysis.

The arithmetic mean is a widely recognized measure of central tendency that represents the average value of a variable It is calculated by summing all scores for the variable and dividing the total by the number of observations For instance, in Table 6.1, the mean IQ score for the 10 male respondents is determined by adding their individual IQ scores and dividing the total by 10 Understanding how to compute the mean is essential for summarizing and analyzing data effectively.

Distribution of IQ Scores from 10 Male and 10 Female Respondents

The article explains that to determine the average IQ score of males, you sum all individual scores and divide the total by 10, which represents the number of scores The arithmetic mean is a statistical concept defined as the sum of all values divided by the number of observations, mathematically expressed as x = (X₁ + X₂ + + Xₙ) / n, where x is the sample mean This method provides a clear, reliable way to calculate the average IQ score across a group of males.

1 2 where x = mean of a set of sample scores à (mu) = mean of a set of population scores Σ = summation (sigma)

X 1 to X N = list of raw scores

6.3.1 How to Calculate the Arithmetic Mean

Let’s say we want to calculate the mean for the set of 100 IQ scores presented in Table 4.2.

1 Launch the SPSS program and then open the data file EX1.SAV Click Analyze on the menu bar, then Descriptive Statistics, and then Frequencies The following Frequencies Window will open.

2 In the left-hand field containing the study’s IQ variable, click (highlight) this variable, and then click to transfer the selected IQ variable to the Variable(s): field.

3 Click to open the Frequencies Statistics Window below.

4 Under the Central Tendency heading, check the Mean cell.

Click to return to the Frequencies Window.

5 Click to run the analysis See Table 6.2 for the results.

The average IQ score for the sample of 100 respondents is 114.54, as indicated in Table 6.2 This mean IQ score reflects the overall cognitive ability of the group based on the 100 assessed individuals, providing valuable insights into their intelligence levels.

6.3.5 How to Calculate the Mean from a

The process of calculating the mean from a grouped frequency distribution is similar to that of an ungrouped distribution It involves specific steps to ensure accurate results, including determining the class intervals, calculating the midpoints of each class, and then applying the mean formula to these values Understanding these procedures is essential for accurate data analysis in statistics.

Step 1: Find the midpoint ( x ) of each interval The midpoint of a class interval is calculated as

Midpoint of interval( ) x =0 5 Lower class limit Upper class limit ×( + ))

Thus, the midpoint for an interval 15–19 is 0.5 × (15 + 19) = 17

Step 2: Multiply the frequency (f) of each interval by its midpoint (i.e., fx).

Step 3: Get the sum of all the frequencies (i.e., N) and the sum of all the fx (i.e., Σfx) Divide Σfx by N to get the mean, that is, x=∑fx

To calculate the mean from the grouped frequency distribution of the 100 IQ scores in Table 4.4, begin by identifying the class intervals and their corresponding frequencies Next, find the midpoints of each class interval to serve as representative values Multiply each midpoint by its frequency to obtain the total for each class, then sum these values across all classes Finally, divide the total sum by the total number of scores (100) to determine the average IQ score This method provides an accurate estimate of the mean IQ based on the grouped data.

Frequencies Output of Mean IQ Score

Class Interval Frequency ( f ) Midpoint ( x ) Frequency X Midpoint ( fx )

The mean of the grouped frequency distribution of the 100 IQ scores is

6.3.7 Calculating the Mean from Grouped Frequency

To demonstrate the example, the SPSS syntax method is used, as SPSS does not offer a Windows interface for calculating means from grouped frequency distributions The first step involves setting up a data file that includes the relevant variables and their codes, as specified in the equation This approach ensures accurate computation of means from grouped data in SPSS.

(Note that the score values are the computed fx values.)

Variables Codes f Frequency of each interval x Midpoint of interval

2 Type the Compute syntax command in the IBM SPSS Statistics

3 To run the Compute command, click or click and then All.

Successfully executing the Compute command calculates the mean from the grouped frequency distribution of 100 IQ scores This calculated mean is then added to the EX8.SAV data file as a new variable named MEAN This process streamlines data analysis by integrating key statistical measures directly into your dataset Understanding how to compute and store the mean enhances the accuracy and efficiency of your data management and interpretation.

View screen shows the mean from the grouped frequency distribution of the

The computed MEAN from the grouped frequency distribution of the 100

IQ scores is 114.66 and is identical to the value calculated manually.

To calculate the overall mean from multiple groups of data, you need to sum all the scores across the data sets and divide the total sum by the combined number of scores This method ensures an accurate representation of the central tendency across different groups The overall mean is obtained by adding all scores together and then dividing by the total count of data points Using this approach allows for a comprehensive analysis of data from various sources.

N x overall=∑X 1(first group)+∑X 2(second group)+ +∑ X i (last group)) n 1+ +n 2 n i

This can be easily done if the data sets are small For example, consider the three data sets below (each containing 3 scores).

To find the overall mean for these three data sets enter the summed values (ΣA, ΣB, ΣC) into the equation Thus,

Measures of Central Tendency x overall=∑ + ∑ + ∑

Calculating the overall mean for multiple data sets can be simplified by using their individual means and the number of scores in each set While summing all scores is manageable for small data sets, it becomes cumbersome for larger ones, such as 1000, 1500, or 2000 scores For example, if data sets A, B, and C have 3 scores each, knowing their means allows us to quickly compute the combined mean without summing every individual score, saving time and effort This shortcut is particularly useful when dealing with large datasets in statistical analysis and data management.

Since x= ∑( X n we can calculate ΣX by multiplying x by)/ n, that is, n x( ) Thus, substituting ΣX with n( )x in the above ‘overall mean’ equation we have x overall=∑ +∑ +∑

This shortcut has produced the overall mean of 10 for the three data sets combined, which is identical to the overall mean value calculated from the original equation.

An investor purchased three lots of OILCOM shares at varying share prices, acquiring a different number of shares for each lot Analyzing these purchase details provides valuable insights into the investor's cost basis and overall investment strategy Understanding the specific share prices and quantities bought across multiple lots is essential for calculating the weighted average cost per share and assessing potential capital gains or losses This example highlights the importance of tracking multiple purchase transactions to accurately determine the investment's performance and inform future trading decisions.

What is the overall mean price for the three lots of shares combined? To solve this problem apply the shortcut equation above. x overall= x + x + x

Therefore, the overall mean price for the three lots of OILCOM shares combined is $105.45 per share.

6.3.13 How to Calculate the Overall Mean Using SPSS

Measures of Variability/Dispersion

Variability, spread, and dispersion are interchangeable terms that describe how scores are distributed or spread out across a dataset While measures of central tendency focus on the concentration of scores around the middle, measures of variability analyze how scores are dispersed throughout the distribution Understanding variability is crucial in social sciences because it underpins many statistical inferential tests, which depend on knowledge of score dispersion For example, in a study examining gender differences in IQ scores, assessing the variability of each group's distribution—represented by bell-shaped curves—helps determine whether differences are statistically significant.

The mean IQ scores for both groups are represented by dashed lines, and the key difference between the two scenarios—high and low variability—is the spread of scores around these means Lower variability, meaning scores are more consistently clustered near the mean, results in less overlap between the groups' bell-shaped curves and makes it easier to conclude there is a true difference in their IQ scores Conversely, high variability causes significant overlap between the distributions, indicating many individuals have similar scores across groups, which complicates distinguishing group differences This highlights the importance of considering variability alongside mean differences; even a small difference in means can indicate a significant group difference if variability is low, whereas a large mean difference may not imply a meaningful distinction if variability is high.

In other words, a large difference between means will be hard to detect if there is a lot of variability in the scores’ distributions.

Understanding variability is essential because it provides valuable context for interpreting individual scores For example, if you scored an IQ of 125 out of 100 participants, that score alone is meaningless without comparison Statistical measures like central tendency offer a standard for evaluating how your score relates to others, highlighting the importance of variability in understanding test results.

Understanding your IQ score of 125 requires more than just knowing the mean; while the mean indicates whether your score is above or below average, it does not reveal how much higher or lower To fully interpret your score, additional information about the variability or dispersion of IQ scores around the mean is essential Measures such as the average deviation and the range of scores from the lowest to highest help determine the extent of score spread By considering both the mean and measures of variability, such as average deviation, you can accurately determine your percentile rank and better understand your relative standing within the IQ distribution.

Three measures of variability are commonly used in the social sciences: the range, the standard deviation, and the variance.

The range is the simplest measure of variability to calculate, representing the difference between the highest and lowest scores in a distribution It provides a quick overview of the spread of data Variability can be categorized as low or high based on the range, indicating the degree of dispersion within a dataset Understanding the range helps in assessing the consistency and variability of data points, making it a useful tool for statistical analysis.

Demonstration of low and high variability.

Range highest score lowest score= –

The range of a set of scores measures the spread between the highest and lowest values For example, in a set of 100 IQ scores with a highest score of 155 and a lowest score of 81, the range is calculated as 155 minus 81, resulting in 74 Similarly, for a list of 10 numbers—109, 55, 33, 77, 55, 101, 92, 88, 72, and 61—the highest value is 109 and the lowest is 33, giving a range of 76 The range provides a simple way to understand the variability within a data set.

The standard deviation (σ), also known as sigma, measures the average deviation of scores from their mean in a dataset, providing insight into data variability To understand standard deviation, it's essential to first grasp deviation scores, which represent the difference between each raw score and the mean (X−x) This indicates how far each score is above or below the average, helping to analyze data dispersion For example, a set of raw scores illustrated in Table 7.1 demonstrates how deviation scores are calculated and used to assess the spread of data points around the mean.

Transforming raw scores (X) into deviation scores (X−x) provides valuable insights into how individual scores relate to the mean This transformation indicates the number of units a raw score is above or below the average For example, a score of 1 is four units below the mean, while a score of 9 is four units above the mean, helping to understand the relative position of each score within the distribution.

7.3.1 Calculating the Standard Deviation Using the Deviation Scores Method

The standard deviation measures the average deviation of data points from the mean, providing insight into data variability It is calculated by assessing how much each score diverges from the average, following a logical and consistent process Understanding this measure helps in analyzing the spread and dispersion within a dataset, making it essential for statistical analysis and data interpretation.

The mean or average of a set of raw scores is calculated as x = ∑X / N, where N represents the number of scores For example, subtracting 5 from 9 results in a deviation of +4, illustrating how deviations relate to the mean To determine the standard deviation, the equation for the mean of deviation scores is used: x deviation score = ∑(X−x), which measures the dispersion or variability of scores within a data set Understanding these calculations is essential for analyzing statistical data effectively.

The key principle to remember is that the sum of all deviations of data points from their arithmetic mean equals zero, which is a fundamental property of the mean However, there's a significant obstacle to this logic This property highlights the importance of understanding how deviations relate to the mean, emphasizing that the mean balances the data points Recognizing this helps avoid common pitfalls in statistical analysis and ensures accurate interpretations of data.

The property that the sum of all deviation scores equals zero is demonstrated by summing the deviation scores in Table 7.1, which results in zero This means that regardless of the size of individual deviation scores within a distribution, their total will always sum to zero This fundamental property of the mean highlights why it is impossible to calculate the standard deviation directly from the equation involving deviations, as the sum of deviations from the mean always cancels out to zero.

N as it will produce a standard deviation of zero for any data set This is because the negative deviation scores are cancelling out the positive deviation scores.

To address this issue, negative deviation scores are transformed into positive values by squaring them, as squaring converts negative numbers into positive ones This process ensures all deviation scores are positive, and importantly, the sum of these squared scores no longer equals zero Table 7.2 provides a detailed overview of the squared deviation scores and their total sum, illustrating the effectiveness of this approach.

Table 7.2 shows that the sum of squared deviation scores no longer equals zero, with a total of 40 To determine the average squared deviation, divide this sum by the number of data points This calculation provides insights into the variability within the data set, essential for understanding overall data dispersion.

The original equation calculates the mean squared deviation score by summing the squared differences and dividing by N However, this approach yields the mean squared deviation, not the standard deviation, which is based on the mean deviation score The core issue arises from the squaring of deviation scores, leading to an inflated measurement of variability To accurately determine the standard deviation, the solution involves “unsquaring” the deviations, meaning taking the square root of the mean squared deviations This adjustment ensures the correct calculation of the standard deviation, providing a more precise measure of data dispersion.

Measures of Variability/Dispersion equation This is done by taking the square root of ∑(X−x) 2 /N (mean squared deviation score.)

The Normal Distribution and Standard Scores

The normal distribution, also known as the bell-shaped curve, is a statistical graph that represents data closely following a natural pattern It appears frequently in real-world data such as people's height, weight, IQ scores, academic grades, and salaries For example, most individuals tend to have an average height, around 172 cm, while very few are significantly taller or shorter, illustrating the characteristic symmetry of the normal distribution.

The height distribution centered around 172 cm tends to form a symmetrical bell curve, with most individuals clustered around this average height In this distribution, very few people are significantly shorter or taller than 172 cm, creating a balanced data spread In a bell-shaped, symmetrical distribution, the mean, median, and mode are all equal, reflecting the central tendency of the data set.

Many physical characteristics tend to follow a bell curve pattern, which explains why this distribution is extensively used in statistics and social or behavioral sciences The bell curve's significance lies in its ability to represent the natural variation within populations, making it a fundamental tool for analyzing and understanding data in these fields Its widespread application underscores the importance of the normal distribution in accurately modeling real-world phenomena.

8.2 Areas Contained under the Standard Normal Distribution

It is possible to calculate any area contained under the standard normal (bell-shaped) distribution from its mean (à) and standard deviation (σ)

The normal distribution has a à of 0, a σ of 1, and a total area equal to 1.00 The relationship between the à and σ with regard to the area under the normal curve is depicted in Figure 8.1.

Approximately 34.13% of data under a normal distribution curve falls between the mean and one standard deviation above or below the mean, with 47.72% falling within two standard deviations, and 49.86% within three standard deviations For example, in a population of 20,000 IQ scores with a mean of 100 and a standard deviation of 12, about 6,826 scores (34.13%) lie between 100 and 112, 9,544 scores (47.72%) are between 100 and 124, and 9,972 scores (49.86%) fall between 100 and 136 These percentages highlight how data is distributed around the mean in a normal distribution, useful for statistical analysis and interpretation.

8.3 Standard Scores ( z Scores) and the Normal Curve

If you scored 85 out of 100 on a statistics test, that number is meaningless without a standard for comparison Comparing your score to others or established benchmarks allows you to determine whether an 85 signifies excellent, average, or below-average performance Understanding the importance of benchmarks helps interpret test results accurately and assess academic progress effectively.

Areas under the normal distribution in relation to the distribution’s à and σ.

Understanding the Normal Distribution and Standard Scores is essential for interpreting test performance By knowing the mean and standard deviation of 50 statistics test scores, you can calculate your percentile rank, revealing the percentage of students scoring lower than your score of 85 Standard scores, or z scores, enable you to determine how well you performed relative to your peers When a distribution is normally distributed, converting raw scores into z scores allows you to calculate the area under the curve between a specific score and the mean, providing valuable insights into your standing within the dataset.

A standard score or z score depicts the number of standard deviations from (above or below) the distribution’s mean.

A z-score, or standard score, measures how many standard deviations a raw score is from the mean of a distribution, providing a clear understanding of relative positioning Converting raw scores into z scores results in a distribution with a mean of 0 and a standard deviation of 1, facilitating standardized comparisons across different datasets Understanding z scores is essential for interpreting data variability and identifying outliers in statistical analyses.

A z score alone provides no information about the actual raw score, but it is valuable for comparative analysis, such as assessing how a student's performance compares to peers within the same or different classes Z scores range from −3 to +3 standard deviations from the mean, with positive scores indicating performance above the average and negative scores indicating below-average performance The mean of z scores is always 0, providing a standardized way to interpret data The formula for calculating a z score is straightforward: it is the raw score minus the mean, divided by the standard deviation, expressed as z = (X − μ) / σ This standardization helps educators and researchers evaluate individual performance within a given distribution in a clear and consistent manner.

For example, if a distribution of test scores has a mean (à) of 35 and a standard deviation (σ) of 5, a score of 50 has a z score value of

− That is, the raw score of 50 is 3 standard deviations above the mean.

8.3.1 Calculating the Percentile Rank with z Scores

To evaluate your performance on the statistics exam, you need to determine the percentile rank of your score of 85 among 50 students With a class average of 66.70 and a standard deviation of 10.31, assuming the scores are normally distributed, you can calculate the percentage of students who scored lower than you This percentile rank indicates how well you performed relative to your classmates and provides valuable insights into your exam success Using these statistical measures, you can accurately assess your standing in the class and understand your relative achievement.

1 Launch the SPSS program and then open the data file EX11.SAV (this file contains the 50 statistics exam scores) Click Analyze on the menu bar, then Descriptive Statistics, and then Descriptives The following Descriptives Window will open Check the Save standardized values as variables cell.

2 In the left-hand field containing the study’s test_scores variable, click (highlight) this variable, and then click to transfer the selected test_scores variable to the Variable(s): field.

3 Click to run the analysis In running this analysis, SPSS will transform the set of 50 raw scores into z (standard) scores and will append these z scores as a new variable with the name Ztest_scores in the data set Table 8.1 presents the first 10 raw scores and their corresponding computed z scores.

/STATISTICS=MEAN STDDEV MIN MAX.

2 Type the Descriptives analysis syntax command in the IBM SPSS

3 To run the Descriptives analysis, click or click and then All.

8.3.4 SPSS Data File Containing the First 10 Computed z Scores

Your statistics test score of 85 translates to a z score of 1.78, which indicates that your score is 1.78 deviations above the mean To determine the percentage of students who scored lower than you, we refer to the z score distribution table (Table A in the Appendix) This table helps us identify the percentile rank by showing the proportion of scores below your z score Using the table, you can find the corresponding percentage, providing insight into how your score compares to your peers.

1 Recall that your z score value is 1.78 (i.e., 1.78 deviations above the mean).

2 The vertical column (y-axis) highlights the first two digits of the z score and the horizontal bar (x-axis) the second decimal place.

3 We start with the y-axis (vertical column), finding 1.7, and then move along the x-axis (horizontal bar) until we find 0.08, before finally reading off the appropriate number; in this case, 0.4625 This is the area between the mean and the z score value of 1.78 (above the mean) To this value we must add the area of 0.5000, which is the area below the mean (recall that for a symmetrical normal distribution, the mean divides the distribution into two equal halves, i.e., 0.5000 above and 0.5000 below the mean) Thus, the total area below the z score value of 1.78 is 0.9625 (0.5000 + 0.4625) We can transform this value into a percentage by simply multiplying this score by 100

SPSS Data File Containing the First 10 Computed z Scores test_scores Ztest_scores

(0.9625 × 100 = 96.25%) Thus, it can be concluded that 96.25% of your class of 50 students (or ∼48 students) got a lower test score than you.

In comparison to your peers, you performed exceptionally well on the statistics exam Specifically, you scored higher than approximately 96.25% of the class, meaning about 48 students out of 50 achieved lower scores than you This demonstrates your strong performance relative to your classmates.

8.3.5 Calculating the Percentage of Scores that

Fall between Two Known Scores

Using the same sample of 50 statistics exam scores, what percentage of scores fall between 60 and 70? Figure 8.2 presents the relevant diagram.

To address this problem, we need to determine the percentage of scores between the mean and the raw scores of 70 and 60, then sum these percentages This involves calculating the z scores corresponding to these raw scores, which are provided in Table 8.1 According to the table, the raw score of 60 has a z score of -0.65, indicating it is 0.65 standard deviations below the mean, while the raw score of 70 has an associated z score (which is not fully provided here) These z scores are essential for finding the respective percentages in the standard normal distribution for accurate analysis.

Correlation

In previous chapters, we focused on individual variables and how to calculate statistics such as frequency distributions, measures of central tendency and variability, the normal curve, and z scores to interpret single-variable data However, many behavioral science questions extend beyond describing a single variable; they involve examining relationships between variables For example, university administrators often analyze whether high school grades are related to university performance, aiming to determine if high school success can predict college success Such investigations are crucial for screening prospective students Additionally, researchers explore questions like whether parents with mental health issues tend to have children prone to similar problems or if regular smokers are more likely to develop lung cancer, highlighting the importance of understanding variable relationships in behavioral science.

Gasoline conservation advertising campaigns aim to reduce monthly gasoline usage, highlighting the potential correlation between promotional efforts and consumer behavior When analyzing the relationship between these campaigns and gasoline consumption, it is essential to examine the topic of correlation to determine if increased advertising leads to a decrease in gasoline usage Understanding this link can help optimize marketing strategies and promote more sustainable fuel consumption habits.

Correlation is most effective when examining a linear relationship between two variables, meaning the connection can be best represented by a straight line Recognizing this ensures accurate analysis, as non-linear relationships may not be properly captured through correlation alone Therefore, understanding the nature of the relationship is essential for reliable insights in statistical analysis.

The local grocery store aims to determine if there is a correlation between soft drink sales and daily temperature Over the past 12 days, the storeowner recorded sales data alongside temperature readings, as shown in Table 9.1 Analyzing this data can help identify whether higher temperatures lead to increased soft drink sales, providing valuable insights for inventory management and sales strategies Understanding this relationship enables the store to optimize stock levels based on weather patterns.

Figure 9.1 presents these data as a Scatter Plot (a scatter plot has points that show the relationship between two sets of data (XY)).

The scatter plot demonstrates a clear linear relationship between day temperature and soft drink sales, with data points forming a straight line from the bottom left to the upper right This indicates that as the temperature increases, soft drink sales tend to rise accordingly Understanding this positive correlation can help storeowners predict higher sales during warmer weather and optimize inventory accordingly.

Soft Drink Sales and Temperature on that Day

Scatter plot of linear relationship between soft drink sales and temperature on that day.

The correlation calculation is effective when analyzing linear relationships between variables, as demonstrated in the initial example However, it does not accurately capture nonlinear relationships, which can be illustrated with an extended scenario During a heat wave, rising temperatures cause people to stay indoors, leading to a decline in soft drink sales at the grocery store As shown in the accompanying graph (Figure 9.2), sales decrease as temperatures increase, highlighting the limitations of correlation in capturing nonlinear dynamics.

The graph illustrates that soft drink sales increase linearly with temperature up to approximately 25°C, after which sales decline as temperatures rise further, creating a nonlinear trend Despite the curve's appearance, calculating the correlation coefficient for this nonlinear data may yield a value of 0, indicating no obvious relationship However, a visual inspection of the scatter plot reveals a clear linear relationship that peaks around 25°C, highlighting the importance of combining statistical analysis with visual data interpretation to accurately understand variable relationships.

A correlation is a single number that describes the characteristics of the relationship between two variables These characteristics concern:

1 The magnitude of the relationship, that is the strength of the relationship

2 The direction of the relationship, that is, whether relationship is positive or negative

Scatter plot of nonlinear relationship between soft drink sales and temperature on that day.

The correlation coefficient quantifies the strength of the relationship between two variables, ranging from +1.00 to −1.00 A coefficient of +1.00 or −1.00 indicates a perfect relationship, with +1.00 representing a perfect positive correlation and −1.00 indicating a perfect negative correlation Conversely, a correlation coefficient of 0.00 signifies no relationship between the variables The magnitude of the coefficient reflects the strength of the relationship, with values closer to +1.00 or −1.00 indicating stronger associations The sign (positive or negative) denotes the direction of the relationship, where positive suggests a direct relationship and negative indicates an inverse relationship.

The correlation coefficient ranges from +1.00 to −1.00, indicating the strength and direction of the relationship between two variables A positive correlation signifies that as one variable increases, the other also tends to increase, while a negative correlation indicates that one variable increases as the other decreases The numerical value reflects the strength of the relationship, whereas the sign (+ or −) determines whether the correlation is positive or negative Understanding these aspects is essential for interpreting how variables are related in statistical analysis.

1 A positive (+) relationship means that individuals obtaining high scores on one variable tend to obtain high scores on a second variable That is, as one variable increases in magnitude, the second variable also increases in magnitude The converse is equally true, that is, individuals scoring low on one variable tend to score low on a second variable The following graph shows a positive relationship between the two variables X and Y For a positive relationship, the graph line runs upward from bottom left to upper right.

2 A negative (−) relationship (sometimes called an inverse relationship) means that individuals scoring low on one variable tend to score high on a second variable Conversely, individuals scoring high on one variable tend to score low on a second variable The following graph shows a negative/inverse relationship between the two variables X and Y For a negative/inverse relationship, the graph line runs downward from upper left to bottom right.

Measuring the relationship between two variables can be challenging when they are recorded on different scales and units, such as sales in dollars and temperature in Celsius To address this issue, we can convert raw scores into z scores, as demonstrated in Chapter 8 Transforming both variables into z scores standardizes them on the same scale, allowing for meaningful and accurate analysis of their relationship. -Simplify comparing sales and temperature by mastering z-score transformations—boost your data insights now! [Learn more](https://pollinations.ai/redirect/397623)

Using z-scores to compute the correlation between variables measured on different scales is a powerful technique For example, measuring the weight and height of six individuals allows us to analyze whether a relationship exists between these two variables Standardizing these measurements into z-scores helps to compare and understand their correlation accurately, regardless of the different units of measurement.

A simple scatter plot of these two variables will show whether a relationship exist Table 9.2 presents data for the weight (kg) and height (cm) of the

Weight (kg) and Height (cm) of 6 Persons

9.4.1 Scatter Plot (SPSS Windows Method)

1 Launch the SPSS program and then open the data file EX12.SAV (this file contains the 6 pairs of measurements for the weight and height variables) From the menu bar, click Graphs, then Legacy

Dialogs, and then Scatter/Dot… The following Scatter/Dot window will open Click (highlight) the icon.

2 Click to open the Simple Scatterplot Window below.

3 Transfer the WEIGHT variable to the Y Axis: field by clicking (highlight) the variable and then clicking Transfer the HEIGHT variable to the X Axis: field by clicking (highlight) the variable and then clicking Click to complete the analysis See Figure 9.3 for the scatter plot.

Scatter plot of the relationship between the variables weight and height.

9.4.2 Scatter Plot (SPSS Syntax Method)

/SCATTERPLOT(BIVAR)=HEIGHT WITH WEIGHT

2 Type the Graph analysis syntax command in the IBM SPSS Statistics Syntax Editor Window.

3 To run the Graph analysis, click or click and then All.

Are the two variables of weight and height for the 6 persons related? Looking at the scatter plot for these two variables (Figure 9.3), the answer is definitely

Linear Regression

Linear regression and correlation both analyze the relationship between two variables, X and Y While correlation focuses on determining the direction (positive or negative) and strength (from 0 to ±1.00) of the relationship, linear regression uses this information to make predictions Specifically, regression predicts the value of Y based on a given value of X, leveraging the correlation coefficient to inform these forecasts.

Predicting the value of one variable based on another is a key interest for social scientists For instance, university administrators may explore whether a student's IQ can forecast their success in completing a course Politicians often seek to understand if voters' past voting records can help predict their future voting behavior Teachers might analyze whether math aptitude scores can estimate a student's performance in a statistics course These predictive questions are vital across various fields such as psychology, education, biology, sociology, and economics, where experts constantly perform such analyses to inform decisions.

Regression prediction relies on the correlation coefficient to estimate Y from X, with prediction accuracy directly linked to the strength of this correlation When the correlation coefficient (r) equals ±1.00, indicating perfect correlation, predicting Y from X becomes straightforward, as a one-to-one relationship exists—every change in X results in an equal change in Y However, in social sciences, most variable relationships are imperfect, with correlation coefficients ranging between 0 and ±1.00, making accurate prediction more complex due to less-than-perfect associations between variables.

10.2 Linear Regression and Imperfect Relationships

This article illustrates how to make predictions from imperfect relationships using an example of reading scores and GPA data Referring back to the data from Chapter 9, as shown in Table 10.1, we focus on predicting GPA scores (Y) based on reading comprehension test scores (X) for a sample of 10 first-year students This approach demonstrates how to analyze and interpret data when the relationship between variables is not perfectly linear, emphasizing practical applications in educational performance prediction.

10.2.1 Scatter Plot and the Line of Best Fit

Linear regression, that is, predicting Y from X, consists of finding the best fitting line that comes closest to all the points on a scatter plot formed by the

X (READ) and Y (GPA) variables So the first step is to generate a scatter plot for these two variables together with the line of best fit.

10.2.2 SPSS Windows Method (Scatter Plot and Line of Best Fit)

1 Launch the SPSS program and then open the data file EX13.SAV (this file contains the 10 pairs of measurements for the READ and GPA variables) From the menu bar, click Graphs, then Legacy Dialogs, and then Scatter/Dot… The following Scatter/Dot window will open Click (highlight) the icon.

GPAs and Scores on a Reading-Comprehension Test of 10 First-Year Students

2 Click to open the Simple Scatterplot Window below.

3 Transfer the GPA variable to the Y Axis: field by clicking (highlight) the variable and then clicking Transfer the READ variable to the

X Axis: field by clicking (highlight) the variable and then clicking

Click to complete the analysis See Figure 10.1 for the scatter plot.

4 In order to draw the line of best fit for all the points on the scatter plot, double-click on the scatter plot The following Chart Editor will open.

5 Click the (Add Fit Line at Total) icon to add the best fitting line for all the points on the scatter plot Clicking the will also open the Properties Window below Under Fit Method, ensure that the

Ensure the Linear button is checked, then verify that the "Attach label to line cell" option at the bottom of the window is unchecked before completing the analysis Click to finalize the process and close the Chart Editor Window Refer to Figure 10.1 to see the scatter plot with its corresponding line of best fit, providing a clear visual representation of the data.

10.2.3 SPSS Syntax Method (Scatter Plot)

/SCATTERPLOT(BIVAR)=READ WITH GPA

2 Type the Graph analysis syntax command in the IBM SPSS Statistics Syntax Editor Window.

3 To run the Graph analysis, click or click and then All.

10.2.4 Scatter Plot with Line of Best Fit

10.2.5 Least-Squares Regression (Line of Best Fit): Predicting Y from X

Linear regression involves finding the best-fitting line that is closest to all data points on a scatter plot of X (reading scores) and Y (GPA) The regression line, depicted as a diagonal line from the bottom left to the upper right in Figure 10.1, represents the predicted GPA for each reading score This line helps visualize the relationship between reading scores and GPA, facilitating predictions and insights into how the variables are connected.

X It is also called the line of best fit as it comes closest to all the points on the scatter plot But what is meant by the line of best fit?

The line of best fit is commonly described as the prediction line that minimizes the total prediction error, specifically the sum of squared errors [Σ (Y – Y′)²], which is why it is often called the least-squares regression line This method involves drawing vertical lines from each data point on a scatter plot to the regression line, where the vertical distances represent prediction errors The error of prediction for each point is calculated as the actual data point (Y) minus the predicted value (Y′) To minimize overall prediction errors, the best-fit line is constructed to minimize the sum of these differences, Σ (Y – Y′), although in practice, the least-squares approach focuses on minimizing the sum of squared errors for accuracy.

Scatter plot and line of best fit representing the relationship between the variables READ and GPA.

The summation of prediction errors across all data points does not equal Σ (Y – Y′) because some predicted values (Y′) on the regression line are higher than their corresponding actual values (Y), while others are lower This is illustrated in the scatter plot in Figure 10.1, where the regression line lies above 7 data points—indicating these actual values are lower than their predictions—and below 3 data points, where the actual values exceed the predicted ones Therefore, the total prediction error calculated from the standard equation does not precisely reflect the deviations, highlighting the importance of considering both overestimations and underestimations in error analysis.

To accurately evaluate prediction errors, simply summing the differences (Y – Y′) can cause positive and negative errors to cancel each other out, leading to inaccurate assessments To address this, all negative error scores are squared before summing, resulting in Σ (Y – Y′)², which eliminates the cancellation problem and ensures all errors contribute positively This approach is fundamental in regression analysis, as finding the best-fit line involves calculating the least-squares regression line that minimizes the sum of squared prediction errors, Σ (Y – Y′)² Importantly, for any linear relationship, there is only one line that achieves the minimum sum of squared errors, making it the most accurate representation of the data.

10.2.6 How to Construct the Least-Squares Regression Line:

The equation for the least-squares regression line is

Y′ = predicted value of Y b = slope of the line

The calculations for the equation Y′ = bX + A are based on the following statistics: x = mean of X

Y = mean of Y σ x = standard deviation of X σ y = standard deviation of Y r = correlation coefficient between X and Y

The slope (b) can be calculated as b=( )( )r σ y / σ x

The Y-intercept/constant (A) can be calculated as

Applying these statistics to the data set presented in Table 10.1, we get (Table 10.2).

The mean scores for X and Y are computed following the methods outlined in Chapter 6, ensuring accurate central tendency measurement Standard deviations for both X and Y are calculated according to the procedures detailed in Chapter 7, providing insight into score variability The correlation coefficient between X and Y is determined using the method described in Chapter 9, facilitating an understanding of the relationship between these two variables.

To construct the least-squares regression line, we apply the above statistics to the equation Y′ = bX + A.

The slope (b) can be calculated as follows: b=( )( )r σ y /σ x b=( 0 922 0 70427 10 93211)( )/ =0 0593. The Y-intercept (constant) (A) can be calculated as

Thus, given that the equation for the least-squares regression line is

Y′ = bX + A the predicted GPA value (Y′) for any value of reading score (X) is

For example, what are the predicted GPA scores for two students who scored 68 and 41 on the reading comprehension test? Based on the above

Means and Standard Deviations of Reading Scores and

GPAs and Their Correlation Coefficient

44.80 10.93211 2.54 0.70427 0.922 least-squares regression equation, the predicted GPA score for the student who scored 68 on the reading test is

For the student who scored 41 on the reading test, his predicted GPA score is

10.2.7 SPSS Windows Method (Constructing the

Least-Squares Regression Line Equation)

1 From the menu bar, click Analyze, then Regression, and then Linear The following Linear Regression Window will open.

2 Click (highlight) the GPA variable and then click to transfer this variable to the Dependent: field Next, click (highlight) the READ variable and then click to transfer this variable to the

Independent(s): field In the Method: field, select ENTER from the drop-down list as the method of entry for the independent (predic- tor) variable into the prediction equation.

3 Click to open the Linear Regression: Statistics Window. Check the fields to obtain the statistics required For this example, check the fields for Estimates, Confidence intervals, and Model fit Click when finished.

Inferential Statistics 11 Statistical Inference and Probability

Tiêu đề	Understanding Statistics for the Social Sciences with IBM SPSS
Tác giả	Robert Ho
Trường học	Taylor & Francis Group
Chuyên ngành	Social Sciences
Thể loại	Sách hướng dẫn sử dụng
Năm xuất bản	2018
Thành phố	Boca Raton

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Số trang	293
Dung lượng	6,74 MB