MATH 1401 · Dr. Barry Monk

Graphical Summaries

Sections 2.1 & 2.2

Section 2.1 · Objective 1

Construct frequency distributions for qualitative data

Qualitative Data

Qualitative data includes categories or labels. For example, consider the types of credit cards used by the last 50 customers at a retailer.

DISCOVERVISAAMEXVISAMASTERCARDVISAVISAMASTERCARDMASTERCARDMASTERCARD VISAVISAMASTERCARDAMEXVISAAMEXVISAAMEXDISCOVERDISCOVER VISAAMEXVISAAMEXDISCOVERDISCOVERVISAVISAMASTERCARDAMEX AMEXMASTERCARDVISAMASTERCARDAMEXVISAVISAMASTERCARDVISADISCOVER VISAVISAVISAVISADISCOVERMASTERCARDMASTERCARDVISAVISAVISA

MasterCard11

Visa23

American Express9

Discover7

Definition

Frequency Distribution

The frequency of a category is the number of times it occurs in the data set.

A frequency distribution is a table that presents the frequency for each category.

Type of Credit Card	Frequency
MasterCard	11
Visa	23
American Express	9
Discover	7

Definition

Relative Frequency

A frequency distribution displays how many observations are in each category. Sometimes, we are interested in the proportion of observations in each category.

The proportion of observations in a category is called the relative frequency of the category.

The relative frequency of a category is the frequency of the category divided by the sum of all frequencies.

Relative Frequency = \frac{Frequency}{Sum of all frequencies}

A tall glass jar packed with brightly colored candy, seen from the side so the mix of colors fills the jar.

Example

Example: Relative Frequency

To construct the relative frequency distribution for the credit card data, we begin by summing the frequencies:

11 + 23 + 9 + 7 = 50

Next, compute the relative frequency for each type of credit card.

Type of Credit Card	Frequency	Relative Frequency
MasterCard	11	11/50 = 0.22
Visa	23	23/50 = 0.46
American Express	9	9/50 = 0.18
Discover	7	7/50 = 0.14

Section 2.1 · Objective 2

Construct bar graphs

Bar Graphs

A bar graph visually represents a frequency distribution using rectangles of equal width, with one rectangle for each category. The height of each rectangle corresponds to the frequency or relative frequency of that category.

Pareto Chart

Sometimes it’s helpful to create a bar graph where categories are ordered by frequency or relative frequency. This type of graph is called a Pareto chart.

Horizontal Bars

In a bar graph, the bars can be either horizontal or vertical. Horizontal bars are often more convenient when the categories have long names.

Side-by-Side Bar Graphs

When comparing two bar graphs with the same categories, it’s best to place both graphs on the same axes, positioning the bars for each category side by side. This makes it easier to see differences between the two sets of data.

2023 2026

Section 2.1 · Objective 3

Construct pie charts

Definition

Pie Charts

A pie chart is an alternative to the bar graph for displaying relative frequency information.

The relative sizes of the slices match the relative frequencies of the categories.

For example, if a category has a relative frequency of 0.25, its slice will cover 25% of the circle.

Example

Example: Pie Chart

The pie chart shows the relative frequencies for the credit card data. It’s customary to label each sector with its relative frequency as a percentage.

Type of Credit Card	Relative Frequency
MasterCard	11/50 = 0.22
Visa	23/50 = 0.46
American Express	9/50 = 0.18
Discover	7/50 = 0.14

Touch a row to light its sector.

Check your understanding 1

The bar graph presents the areas of the six largest islands in the world.

Which island is the largest in the world?

Greenland

Is Madagascar and Baffin Island together larger than New Guinea?

Yes

Approximately how large is Borneo?

750,000 square km

Check your understanding 2

A poll asked a sample of people the following question: Do you think things in the United States 5 years from now will be better, worse, or about the same as they are today?

Which was the most common response?

Better

What percentage of people said that things would be the same or worse in 5 years?

57%

Recap · Section 2.1

Same 50 customers, four pictures

Bar graph

Pareto chart

Horizontal bars

Pie chart

Activity

Build a frequency distribution

Category	Freq.	Rel. freq.

Heavy traffic filling every lane of a multi-lane urban expressway, with a hazy city skyline behind it.

Section 2.2 · Objective 1

Construct frequency distributions for quantitative data

Definition

Frequency Distribution for Quantitative Data

To summarize quantitative data, we use a frequency distribution, like qualitative data. However, since quantitative data lack natural categories, we divide them into classes. Classes are intervals of equal width that cover all observed values in the data set.

Lower Class
Limits

Class
0 – 4
5 – 9
10 – 14
15 – 19

Upper Class
Limits

Frequency
2
4
9
3

Class Width = 5 – 0 = 5

Introduction: Air Pollution

How much air pollution is caused by motor vehicles? This question was addressed in a study by Dr. Janet Yanowitz at the Colorado School of Mines.

She studied the emissions of particulate matter, a form of pollution consisting of tiny particles, that has been associated with respiratory disease.

Example

Example: Frequency Distribution 1

The emissions for 65 vehicles, in units of grams of particles per gallon of fuel, are given.

Construct a frequency distribution using a class width of 1.

Example

Example: Frequency Distribution 3

Since the smallest value in the data set is 0.25, we choose 0.00 as the lower limit for the first class. The classes are then constructed using a class width of 1.

Class Limits
0.00 – 0.99
1.00 – 1.99
2.00 – 2.99
3.00 – 3.99
4.00 – 4.99
5.00 – 5.99
6.00 – 6.99

The class width is 1, so the difference in lower limits is 1. The largest data value is 6.64, so every data value is contained in a class.

Example

Example: Frequency Distribution 6

We count the number of observations in each class to obtain the frequency distribution.

Class Limits	Frequency
0.00 – 0.99	9
1.00 – 1.99	26
2.00 – 2.99	11
3.00 – 3.99	13
4.00 – 4.99	3
5.00 – 5.99	1
6.00 – 6.99	2

Relative Frequency Distribution

Like qualitative data, a relative frequency is found by dividing the class frequency by the total frequency.

Relative frequency = \frac{Class frequency}{Total frequency}

Class Limits	Frequency	Relative Frequency
0.00 – 0.99	9	0.138
1.00 – 1.99	26	0.400
2.00 – 2.99	11	0.169
3.00 – 3.99	13	0.200
4.00 – 4.99	3	0.046
5.00 – 5.99	1	0.015
6.00 – 6.99	2	0.031

Section 2.2 · Objective 2

Construct histograms

Histogram

Once a frequency or relative frequency distribution has been created, the information can be put in graphical form by constructing a histogram. Unlike a bar graph, a histogram's horizontal axis is a number line: the classes have a fixed order and the bars touch.

Example

Example: Histogram

The frequency histogram and relative frequency histogram are given for the particulate emissions data.

Note that the two histograms have the same shape. The only difference is the scale on the vertical axis.

Number of Classes

The number of classes can affect the shape of the histogram. Too many classes produce a histogram with too much detail so that the main features of the data are obscured. Too few classes produce a histogram lacking in detail.

The jagged appearance distracts from the overall shape of the data.

12 classesclass width 0.60

Only the most basic features of the data are visible.

Same 65 emissions values throughout — only the grouping changes.

Section 2.2 · Objective 3

Determine the shape of a distribution from a histogram

Shape of a Data Set – Skewed Histograms

A histogram gives a visual impression of the “shape” of a data set. Statisticians have developed terminology to describe some of the commonly observed shapes. A histogram is skewed if one side, or tail, is longer than the other.

A histogram with a long right-hand tail is said to be skewed to the right, or positively skewed.

A histogram with a long left-hand tail is said to be skewed to the left, or negatively skewed.

Discussion

Would you prefer for the course grade distribution in this class to be skewed to the left or skewed to the right?

Skewed to the left — the tall bars sit at the high grades, with only a short tail trailing down toward the low ones.

Shape of a Data Set – Symmetric Histograms

A histogram is symmetric if its right half is a mirror image of its left half. Very few histograms are perfectly symmetric, but many are approximately symmetric.

A symmetric histogram with a peak in the middle is said to be bell-shaped.

A symmetric histogram in which all classes have approximately equal frequencies is said to be uniformly distributed.

Unimodal and Bimodal Histograms

A peak, or high point, of a histogram is referred to as a mode. A histogram is unimodal if it has only one mode, and bimodal if it has two clearly distinct modes.

Unimodal

Bimodal

Check your understanding 4

Classify each of the following histograms as skewed to the left, skewed to the right, approximately bell-shaped, or approximately uniformly distributed.

aSkewed to the right

bSkewed to the left

cApproximately uniformly distributed

dApproximately bell-shaped

Check your understanding 5

Classify each of the following histograms as unimodal or bimodal.

aUnimodal

bUnimodal

cBimodal

Recap · Section 2.2

The named shapes

Skewed to the right

Skewed to the left

Bell-shaped

Uniformly distributed

Unimodal

Bimodal

Discussion 1

Naturally occurring data follows Benford’s Law.

Discussion 2

Three corporations submitted the following expenditures with their taxes.

Corporation #1

79,386	17,988
203,374	80,535
11,967	3,037
100,229	132,056
46,428	59,727
7,012	38,354
957,559	137,648
551,284	4,163
97,439	1,279
780,216	21,404
22,443	323,547
1,023	194,288
238,527	24,346
634,814	695,236
850,840	160,546

Corporation #2

1,393	865,648
47,689	601,981
75,854	262,971
5,395	65,407
53,079	6,892
3,791	748,151
93,401	45,054
129,906	83,821
568,823	228,976
4,693	913,337
21,902	252,378
437,122	82,581
162,182	338,342
7,942	99,613
31,121	78,175

Corporation #3

64,888	374,242
1,643	12,338
832,618	14,204
126,811	31,484
13,545	7,818
2,332	104,625
29,288	44,178
81,074	3,684
401,437	71,665
3,040	15,376
244,676	541,894
49,273	65,928
112,111	250,601
56,776	650,316
262,359	90,852

Activity

Benford’s Law, corporation by corporation

Benford’s Law

Pick a corporation.

A cartoon Uncle Sam pointing directly at the viewer.

Don’t Cheat on Your Taxes!

Sections 2.1 & 2.2

You Are Ready For:

Sections 2.1 & 2.2 HW: Graphical Summaries

Section 2.1

Objective 1Construct frequency distributions for qualitative data

Objective 2Construct bar graphs

Objective 3Construct pie charts

Section 2.2

Objective 1Construct frequency distributions for quantitative data

Objective 2Construct histograms

Objective 3Determine the shape of a distribution from a histogram

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