Chap 2: Graphical descriptive methods

2.1 Types of DataData at least for purposes of Statistics fall into three main groups: • Numerical interval or quantitative data • Nominal categorical or qualitative data • Ordinal ranke

Chapter 2

Graphical descriptive methods

Introduction and Re-cap…

Descriptive statistics

involves arranging, summarising, and presenting a

set of data in such a way that useful information is

Populations and Samples

The graphical and tabular methods presented here

apply to both entire populations and samples drawn

A variable is some characteristic of a population

or sample

E.g student grades.

Typically denoted with a capital letter: X, Y, Z…

The values of the variable are the range of

possible values for a variable

E.g student marks (0…100)

Data are the observed values of a variable.

2.1 Types of Data

Data (at least for purposes of Statistics) fall into three main groups:

• Numerical (interval or quantitative) data

• Nominal (categorical or qualitative) data

• Ordinal (ranked) data

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Numerical Data…

Numerical data

• Real numbers, i.e heights, weights, prices,

waiting time at a medical practice, etc

• Also referred to as quantitative or interval.

• Arithmetic operations can be performed on

numerical data, thus its meaningful to talk

about 2*Height, or Price + $1, and so on

Nominal Data

•The values of nominal data are categories.

E.g responses to questions about marital status are

categories, coded as: Single = 1, Married = 2,

Ordinal Data

•Ordinal data appear to be categorical in

nature, but their values have an order; a

excellent > poor or fair < very goodThat is, order is maintained no matter what

Ordinal Data…

Types of data – Examples

exam grade

HD D C P F

exam grade

HD D C P F

Ordinal data

Food quality

Excellent Good Satisfactory Poor

Food quality Excellent Good Satisfactory Poor

With ordinal data, all we can use is computations involving the ordering

process

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Calculations for Types of Data

As mentioned above,

• All calculations are permitted on interval data

• No calculations are allowed for nominal data,

except counting the number of observations in each category and calculating their proportions

• Only calculations involving a ranking process

are allowed for ordinal data.

This lends itself to the following ‘hierarchy of

data’…

Hierarchy of Data…

Numerical

• Values are real numbers.

• All calculations are valid.

• Data may be treated as ordinal or nominal.

• Values must represent the ranked order of the data.

• Calculations based on an ordering process are valid.

• Data may be treated as nominal but not as numerical.

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Other Forms of Data

• Cross-sectional data is collected at a certain

point in time across a number of units of interest – marketing survey (observe preferences by

gender, age) – test score in a statistics course exam

– starting salaries of graduates of an MBA

program in a particular year.

• Time-series data is collected over successive

points in time

– weekly closing price of gold

– monthly tourist arrivals in Australia.

2.2 Graphical and tabular

techniques for nominal data

The only allowable calculation on nominal data is

to count the frequency of each value of the

variable

We can summarise the data in a table that

presents the categories and their counts called a

frequency distribution.

A relative frequency distribution lists the

categories and the proportion with which each occurs

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Introduction

• The methods presented apply to both

– the entire population, and

– a sample selected from the population

Graphical techniques

for nominal data

are used primarily for nominal data.

when the raw data can be naturally

categorised in a meaningful manner.

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Bar charts

data.

frequency of each category as a bar rising vertically from the horizontal axis

frequency of the corresponding category.

• Another useful chart to present nominal data is the pie chart.

represent the proportions of appearance for nominal data.

slices whose areas are proportional to the

frequencies (or relative frequencies),

thereby displaying the proportion of

occurrences of each category.

Pie charts

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Example 2.1

• To determine the approximate market share of various women’s magazines in New Zealand, a women’s magazine readership survey was

conducted using a sample of 200 readers

• Data was collected and the count of the

occurrences (frequencies) was recorded for each magazine

• The frequencies were presented in a bar chart

• Then the frequencies were converted to

proportions and the results were presented in a

pie chart

Example 2.1

1 = Australian Women’s Weekly (NZ Edition); 2 = Next;

3 = NZ New Idea; 4 = NZ Woman’s Day; 5 = NZ Women’s Weekly; and 6 = That’s Life.

Australian Women’s Week ly, NZ Edn (1) 36 18

Example 2.1 cont (Excel representation)

The size of each slice in a pie chart is proportional

to the percentage corresponding to the category it

represents

(10/100)(360 0 ) = 36 0

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– Use bar charts also when the order in which

data are presented is meaningful

Trend in total exports, Australia, 1992–2009

2.3 Graphical Techniques for

Numerical Data

There are several graphical methods that are

used when the data are numerical (i.e

quantitative, non-categorical)

The most important of these graphical methods

is the histogram.

The histogram is not only a powerful graphical

technique used to summarise interval data, but

it is also used to help explain probabilities.

Example 2.5

• Providing information concerning the

monthly bills of new subscribers in the first month after signing on with a

telephone company

– collect data

– prepare a frequency distribution

– draw a histogram

As part of a larger study, a long-distance company wanted to acquire information about the monthly bills of new subscribers in the first month after signing with the company The company’s marketing manager conducted a survey of 200 new residential subscribers wherein the first month’s bills were recorded These data are stored

in file XM02-05 The general manager planned to present his findings to senior executives What information can be extracted from these data?

Example 2.5 …

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In Example 2.1 we created a frequency distribution

of the 6 categories In this example we also create

a frequency distribution by counting the number of observations that fall into a series of intervals,

called classes

The justification for the classes chosen below will

be discussed later

Example 2.5 …

We have chosen eight classes defined in such a way that each observation falls into one and only one class These classes are defined as follows:

Classes

Amounts that are less than or equal to 15

Amounts that are more than 15 but less than or equal to 30

Amounts that are more than 30 but less than or equal to 45

Amounts that are more than 45 but less than or equal to 60

Amounts that are more than 60 but less than or equal to 75

Amounts that are more than 75 but less than or equal to 90

Amounts that are more than 90 but less than or equal to 105

Amounts that are more than 105 but less than or equal to 120

Example 2.5 …

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About half (71+37=108)

of the bills are ‘small’,

i.e less than $30.

There are only a few telephone bills in the middle range.

(18+28+14=60)÷200 = 30% i.e nearly a third of the phone bills are $90 or more.

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Building a Histogram…

1) Collect the data

2) Create a frequency distribution for the data…

Class width

widths, but sometimes unequal class

widths are called for.

frequency associated with some classes is too low Then,

– several classes are combined together to form

a wider and ‘more populated’ class

– it is possible to form an open-ended class at the higher or lower end of the histogram.

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Building a Histogram…

1) Collect the data

2) Create a frequency distribution for the data…

How?

a) Determine the number of classes to use [8]

b) Determine how wide to make each class

(assuming equal class width) How?

Look at the range of the data, that is,

Range = Largest observation

– Smallest observationRange = $119.63 – $0 = $119.63

Then each class width becomes:

Building a Histogram…

Building a Histogram…

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Building a Histogram…

Symmetry

A histogram is said to be symmetric if, when

we draw a vertical line down the center of the histogram, the two sides are identical in shape and size:

A skewed histogram is one with a long tail

extending to either the right or the left:

Modality

A unimodal histogram is one with a single peak, while

a bimodal histogram is one with two peaks:

A modal class is the class with

the largest number of observations

Shapes of Histograms…

Bell Shape

A special type of symmetric unimodal

histogram is one that is bell shaped:

Many statistical techniques require

that the population be bell

shaped.

Drawing the histogram helps

Shapes of Histograms…

Compare and contrast the following histograms based

on data from Example 2.7: The marks from the

computer-based statistics course and the manual statistics course have very different histograms…

• Retains information about individual observations that would normally be lost in the creation of a histogram.

• Split each observation into two parts, a stem and a leaf:

• e.g Observation value: 42.19

• There are several ways to split it up…

• We could split it at the decimal point:

• Or split it at the ‘tens’ position (while rounding to the nearest integer in the ‘ones’ position).

• Continue this process for all the observations

Then, use the ‘stems’ for the classes and each

leaf becomes part of the histogram (based on

Example 2.5 data) as follows…

Thus, we still have access to

our original data point’s value!

Stem & Leaf Display…

Histogram and Stem and Leaf

Relative frequency

• It is often preferable to show the

relative frequency (proportion) of

observations falling into each class,

rather than the frequency itself.

Class relative frequency = Class frequency

Total number of observations

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• Relative frequencies should be used

when

– the population relative frequencies are

studied

– comparing two or more histograms

– the number of observations of the samples studied are different

Relative frequency

Cumulative frequency of a class

than the upper limit of that class.

class, we add the frequency of that class

and the frequencies of all previous classes.

particular class is the proportion of

measurements that are less than the upper limit of that class.

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• (pronounced ‘Oh-jive’) is a graph of

a cumulative frequency distribution.

• We create an ogive in three steps…

• First, from the frequency distribution created

earlier, calculate relative frequencies:

Class relative frequency = Class frequency

Total number of observations

• Is a graph of a cumulative frequency

distribution

• We create an ogive in three steps…

 1) Calculate relative frequencies 

 2) Calculate cumulative relative

frequencies by adding the current

class’ relative frequency to the previous

class’ cumulative relative frequency

(For the first class, its cumulative relative

frequency is just its relative frequency.)

TABLE 2.15 Cumulative relative frequencies for Example 2.5

• Is a graph of a cumulative frequency

distribution

• 1) Calculate relative frequencies 

• 2) Calculate cumulative relative

frequencies 

• 3) Graph the cumulative relative

frequencies… Example 2.5 Ogive

Ogive…

The ogive can be used to

answer questions like:

What telephone bill value

is at the 50th percentile?

(Refer also to Fig 2.21 in your textbook.)

around $35

Example 2.5 Ogive

2.4 Describing Time Series Data

• Observations measured at the same point in

time across individual units are called

cross-sectional data.

• Observations measured at successive points in

time on a single unit are called time-series

data

• Time-series data are graphed on a line chart,

which plots the value of the variable on the vertical axis against the time periods on the horizontal axis

• Time series data graphed on a line chart is

Time Series Data

We recorded the value of Australian exports from

1992 to 2009 (Figure 2.22) Draw a line chart to describe these data and briefly describe the

results

Line Chart

• Plot the frequency of a category above the point on the horizontal axis

representing that category.

• Use line charts when the categories are points in time.

• Line charts are particularly useful when the trend over time is to be

emphasised.

Line Chart

Figure 2.22 Line chart showing change in Australian exports over time

Line Chart

• Australian exports have had a slow but steady increase from 1992 to 2004

• After 2004, Australian exports have

been increasing steadily at a much

higher rate.

techniques.

A classification table (or

cross-tabulation table) is used to describe the

A cross-classification table lists the

frequency of each combination of the

values of the two variables…

Describing the Relationship between Two Nominal Variables

Example 2.8

competing newspapers: N1, N2, N3 and N4

advertising managers of the newspapers

need to know which segments of the

newspaper market are reading their papers

relationship between newspapers read and occupation

A sample of newspaper readers was

asked to report which newspaper they

read: N1, N2, N3, N4, and to indicate

whether they were blue-collar worker (1), white-collar worker (2), or professional

(3)

The responses are stored in file XM02-08.

Example 2.8

Example 2.8

the 12 combinations occurs, we produced the Table 2.16.

Example 2.8

then there will be differences in the

newspapers read among the occupations

frequencies in each row to relative

frequencies in each row

Example 2.8

• Interpretation: The relative frequencies in the rows

2 and 3 are similar, but there are large differences between rows 1 and 2, and between rows 1 and 3

• Row 1: Blue collar (1); Row 2: White collar (2);

Row 3: Professional (3)

• This tells us that blue collar workers tend to read different newspapers from both white collar

workers and professionals and that white collar

and professionals are quite similar in their

newspaper choice

newspaper N2 more than twice

as often as newspaper N3.

Describing the Relationship

between Two Numerical Variables

• Often we are interested in the relationships

between two numerical variables

Example 2.9

• A small-business owner wants to assess the effects of advertising on sales levels

• Paired observation data were collected

• Each pair consisted of monthly advertising

expenditure and monthly sales levels

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• A scatter diagram can describe the

relationship between advertising

expenditure and sales.

0 10 20 30 40 50 60

Patterns of Scatter Diagrams…

Linearity and direction are two concepts we are interested in

Weak or non-linear relationship

• That is, they believed that when the price of oil increased the price of petrol also increased, but

Chapter-Opening Example

WERE OIL COMPANIES GOUGING CUSTOMERS 2006?: SOLUTION

1999-• To determine whether this perception is

accurate we determined the monthly figures for both commodities CH02:\Oil

• Graphically depict these data and describe the findings

Chapter-Opening Example

Chapter-Opening Example

Interpreting the results:

• The scatter diagram reveals that the two

prices are strongly related linearly

• As the oil price increases, petrol price also

increases When the price of oil was below

A$85, the relationship between the two

variables was stronger than when the price of oil exceeded A$85