Bài giảng nguyên lý thông kê chương 2 summarizing data student

Why Use Frequency Distributions? A frequency distribution is a way to summarize data  The distribution condenses the raw data into a more useful form.... Simple frequency distributio

Chapter 2 Summarizing data

Summarizing Qualitative Data

Summarizing Quantitative Data

Part A

Why summarizing data?

I Summarizing Qualitative Data

 Frequency Distribution

 Relative Frequency Distribution

 Percent Frequency Distribution

 Bar Graph

 Pie Chart

Example: Marada Inn

Guests staying at Marada Inn were

asked to rate the quality of their

accommodations as being excellent,

above average, average, below average, or

poor The ratings provided by a sample of 20 guests are:

Example: Marada Inn

Excellent Above Average Average

Above Average Above Average Below Average Poor

Above Average Average

Why Use Frequency Distributions?

 A frequency distribution is a way to

summarize data

 The distribution condenses the raw data

into a more useful form

 and allows for a quick visual interpretation

of the data

The relative frequency of a class is

The relative frequency of a class is

A relative frequency distribution is

A relative frequency distribution is

Relative Frequency Distribution

Relative Frequency Distribution

Ratings Frequency Relative frequency

Bar Graph

 A graphical device for depicting qualitative data

 On one axis (usually the horizontal axis), we specify

the label or the name of each of the class

 On the other axis (usually the vertical axis), we specify the frequency, relative frequency of each class

 A bar of fixed width is drawn above each class

label, we extend the height appropriately

 The bars are separated to emphasize the fact that each class is a separate category

 Bar Graph - Example

First draw a circle; then use the relative

frequencies to subdivide the circle

into sectors that correspond to the

relative frequency for each class

 Pie Chart - Example

Insights Gained from the Preceding Pie Chart

Example: Marada Inn

II Summarizing Quantitative Data

1 Simple frequency distribution

 Simple frequency

distribution consists of a

list of data values, each

showing the number of

items having that value

1 Simple frequency distribution

Why?

• Normally not suitable

for continuous data

• Normally applicable

to discrete raw data

Why?

 The following data record the number of

children in the families of the 47 workers in a company:

1 1 3 2 0 2 0 1 2 2 1 3

5 2 4 0 0 2 4 1 1 2 2 0

3 0 0 2 1 3 6 0 2 1 0 3

2 2 2 1 0 0 1 1 3 1 4

Constructing a simple frequency distribution using a tally chart

Data value Tally marks Total

0 1 2 3 4 5 6

Frequency distribution table

Number of children

in family Number of workers

0 1 2 3 4 5 6

Disadvantage of simple frequency

distribution

2 Grouped frequency distributions

 Used when the data set contains a large number

of data values

 A grouped frequency distribution summaries

data into groups of values, each showing the

number of items having values in the group.

 Each group of data value called class

 Used for both continuous data and discrete data

Definitions associated with frequency

distribution classes

 Class limits: are the lower and upper

values of the classes as physically

described in the distribution

Definitions associated with frequency

distribution classes

 Class widths (class lengths):

- continuous data: are the numerical differences between lower and upper class limits.

- discrete data: are the numerical differences

between the lower limit of one class and the lower limit of the immediately following class

 Class mid-points: are situated in the centre of the classes.

Definitions associated with frequency

distribution classes

 Open-ended class:

- A class without a/an

lower/upper limit.

- Usually used for the first

class which has no defined

lower limit and/or the last

class which has no defined

upper limit

Classes

< 10 10-15 15-20

>=20

Guidelines for grouping values into

classes

• Use between 5 and 20 classes

 Number of Data Points Number of Classes

Example: Hudson Auto Repair

The manager of Hudson Auto

would like to have a better

understanding of the cost

of parts used in the engine

tune-ups performed in the

shop She examines 50

customer invoices for tune-ups The costs of parts,rounded to the nearest dollar, are listed on the nextslide

Example: Hudson Auto Repair

Sample of Parts Cost for 50 Tune-ups

Frequency Distribution

 Guidelines for Selecting Width of Classes

Largest Data Value Smallest Data Value

Number of Classes

−

•Use classes of equal width

•Approximate Class Width =

 Example

Relative Frequency Distribution

Part costs ($) Frequency Relative

frequency frequency (%)Relative

Total

Insights Gained from the Percent Frequency Distribution

Relative Frequency Distributions

Dot Plot

 One of the simplest graphical

summaries of data is a dot plot.

 A horizontal axis shows the range of

data values.

 Then each data value is represented by

a dot placed above the axis.

Dot Plot

Tune-up Parts Cost

 Another common graphical presentation of

quantitative data is a histogram

 The variable of interest is placed on the horizontal axis

 A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency

 Unlike a bar graph, a histogram has no gap between rectangle

Tune-up Parts Cost

Cumulative frequency distribution

Cumulative frequency distribution

Cumulative relative frequency distribution –

Cumulative relative frequency distribution –

Cumulative Distributions

Cumulative Frequency Distribution

Part costs ($) Frequency Cumulative

frequency cumulative Relative

frequency

Hudson Auto Repair

An ogive is a graph of a cumulative distribution.The data values are shown on the horizontal axis.Shown on the vertical axis are the:

• cumulative frequencies, or

• cumulative relative frequencies, or

The frequency (one of the above) of each class is plotted as a point

The plotted points are connected by straight lines

Hudson Auto Repair

Ogive with Cumulative Percent Frequencies

Chapter 2 Summarizing data

 Exploratory Data Analysis

 Crosstabulation and Scatter Diagrams

Part B

x y

Exploratory Data Analysis

 The techniques of exploratory data analysis consist of simple arithmetic and easy-to-draw pictures that can

be used to summarize data quickly

 One such technique is the stem-and-leaf display

Stem-and-Leaf Display

Example: Hudson Auto Repair

The manager of Hudson Auto

would like to have a better

understanding of the cost

of parts used in the engine

tune-ups performed in the

shop She examines 50

customer invoices for tune-ups The costs of parts,rounded to the nearest dollar, are listed on the

next

slide

Example: Hudson Auto Repair

Sample of Parts Cost for 50 Tune-ups

Stretched Stem-and-Leaf Display

 Whenever a stem value is stated twice, the first value corresponds to leaf values of 0 - 4, and the second value corresponds to leaf values of 5 - 9

 If we believe the original stem-and-leaf display has condensed the data too much, we can stretch the display by using two stems for each leading digit(s)

Stretched Stem-and-Leaf Display

Stem-and-Leaf Display

Leaf Units

• Where the leaf unit is not shown, it is assumed

to equal 1

• Leaf units may be 100, 10, 1, 0.1, and so on

• In the preceding example, the leaf unit was 1

• A single digit is used to define each leaf

Example: Leaf Unit = 0.1

If we have data with values such as

8.6 11.7 9.4 9.1 10.2 11.0 8.8

a stem-and-leaf display of these data will be

Example: Leaf Unit = 10

If we have data with values such as

1806 1717 1974 1791 1682 1910 1838

a stem-and-leaf display of these data will be

Crosstabulations and Scatter

Diagrams

 Crosstabulation and a scatter diagram are two

methods for summarizing the data for two (or more) variables simultaneously

 Often a manager is interested in tabular and

graphical methods that will help understand the

relationship between two variables

 So far we have focused on methods that are used

to summarize the data for one variable at a time

Crosstabulation can be used when:

 A crosstabulation is

Cross-tabulation Example: Finger Lakes Homes

The number of Finger Lakes homes sold for each style and price for the past two years is shown below

 Insights Gained from Preceding Crosstabulation

Frequency distribution

for the home style variable

Cross-tabulation: Row or Column

Percentages

 Converting the entries in the table into row percentages or column percentages can

provide additional insight about the

relationship between the two variables.

Crosstabulation: Row Percentages

Crosstabulation: Column Percentages

 The general pattern of the plotted points suggests the overall relationship between the variables.

 One variable is shown on the horizontal axis and the other variable is shown on the vertical axis

 A scatter diagram is a graphical presentation of the relationship between two quantitative variables

Scatter Diagram and Trendline

 A trendline is an approximation of the relationship

Scatter Diagram

 A Positive Relationship

x y

Scatter Diagram

 A Negative Relationship

x y

Scatter Diagram

 No Apparent Relationship

x y

Example: Panthers Football Team

 Scatter Diagram

The Panthers football team is interested

in investigating the relationship, if any,

between interceptions made and points scored

13213

1424181730

x = Number ofInterceptions

y = Number of Points Scored

Scatter Diagram

Insights Gained from the Preceding Scatter Diagram

Example: Panthers Football Team

Tabular and Graphical Procedures

Qualitative Data Quantitative Data