Chapter 2

Frequency Distributions

and Graphs

Outline 2-1 Organizing Data 2-2 Histograms, Frequency Polygons, and Ogives 2-3 Other Types of Graphs

Organize data using a frequency distribution. Represent data in frequency distributions

graphically, using histograms, frequency polygons and ogives.

Represent data using bar graphs, Pareto charts, time series graphs, pie graphs, and dotplots.

Draw and interpret a stem and leaf plot.

Objectives

A frequency distribution is the organization of raw data in table form, using classes and frequencies.

Each raw data value is placed into a quantitative or qualitative category called a class.

The frequency of a class is the number of data values contained in a specific class.

1. Categorical

2. Grouped

3. Ungrouped

Twenty-five army inductees were given a blood test to determine their blood type. The data set is

A B B AB OO O B AB BB B O A OA O O O ABAB A O B A

Construct a frequency distribution for the data.

Solution:

PercentFrequencyClass

20%5A

28%7B

36%9O

16%4AB

100%25Total

These data represent the record high temperatures in degrees Fahrenheit for each of the 50 states. Construct a grouped frequency distribution for the data, using 7 classes.

112 100 127 120 134 118 105 110 109 112

110 118 117 116 118 122 114 114 105 109

107 112 114 115 118 117 118 122 106 110

116 108 110 121 113 120 119 111 104 111

120 113 120 117 105 110 118 112 114 114

Solution:

R = H - L = 134 – 100 = 34

Class width = R / number of classes = 34 / 7 = 4.9 ≈ 5

FrequencyClass BoundariesClass Limits

299.5 – 104.5100 – 104

8104.5 – 109.5105 – 109

18109.5 – 114.5110 – 114

13114.5 – 119.5115 – 119

7119.5 – 124.5120 – 124

1124.5 – 129.5125 – 129

1129.5 – 134.5130 – 134

50Total

A cumulative frequency distribution is a distribution that shows the number of data values less than or equal to a specific value (usually an upper boundary).

Cumulative Frequency

0Less than 99.5

2Less than 104.5

10Less than 109.5

28Less than 114.5

41Less than 119.5

48Less than 124.5

49Less than 129.5

50Less than 134.5

The data represent the number of miles per gallon that 30 selected four-wheel-drive sport utility vehicles obtained in city driving. Construct a frequency distribution, and analyze the distribution.

12 17 12 14 16 18

16 18 12 16 17 15

15 16 12 15 16 16

12 14 15 12 15 15

19 13 16 18 16 14

Solution:

R = H - L = 19 – 12 = 7

FrequencyClass BoundariesClass Limits

611.5 – 12.512

112.5 – 13.513

313.5 – 14.514

614.5 – 15.515

815.5 – 16.516

216.5 – 17.517

317.5 – 18.518

118.5 – 19.519

30Total

Cumulative frequency distribution

Cumulative Frequency

0Less than 99.5

2Less than 104.5

10Less than 109.5

28Less than 114.5

41Less than 119.5

48Less than 124.5

49Less than 129.5

50Less than 134.5

The histogram is a graph that displays the data by using contiguous vertical bars (unless the frequency of a class is 0) of various heights to represent the frequencies of the classes.

Construct a histogram to represent the data shown for the record high temperatures for each of the 50 states (see Example 2-2).

The frequency polygon is a graph that displays the data by using lines that connect points plotted for the frequencies at the midpoints of the classes. The frequencies are represented by the heights to of the points.

Using the frequency distribution given in Example 2-4, construct a frequency polygon.

Solution:

FrequencyMidpointsClass Boundaries

210299.5 – 104.5

8107104.5 – 109.5

18112109.5 – 114.5

13117114.5 – 119.5

7122119.5 – 124.5

1127124.5 – 129.5

1132129.5 – 134.5

50Total

The ogive is a graph that displays the cumulative frequencies for the classes in a frequency distribution.

Construct an ogive for the frequency distribution described in Example 2-4.

Solution:

Cumulative Frequency

0Less than 99.5

2Less than 104.5

10Less than 109.5

28Less than 114.5

41Less than 119.5

48Less than 124.5

49Less than 129.5

50Less than 134.5

A bar graph represents the data by using vertical or horizontal bars whose heights or lengths represent the frequencies of the data.

The table shows the average money spent by first-year college students. Draw a horizontal and vertical bar graph for the data.

Electronics $728

Dorm Decor $344

Clothing $141

Shoes $72

Solution:

A pareto chart is used to represent a frequency distribution for a categorical variable, and the frequencies are displayed by the heights of vertical bars, which are arranged in order from highest to lowest.

The data shown here consist of the number of homeless people for a sample of selected cities. Draw and analyze a pareto chart for the data.

Solution:

• The purpose of the pie graph is to show the relationship of the parts to the whole.

• A pie graph is a circle that is divided into sections or wedges according to the percentages in each category of the distribution.

This frequency distribution shows the number of pounds each snack food eaten during the super bowl. Construct a pie graph for the data.

Solution:Degrees = f/n * 360o

Potato chips 11.2/30*360o=134o

Tortilla Chips 8.2/30*360o=98o

Pretzels 4.3/30*360o=52o

Popcorn 3.8/30*360o=46o

Snack nuts 2.5/30*360o=30o

% = f/n * 100

Potato chips 11.2/30*100=37.3%

Tortilla Chips 8.2/30*100=27.3%

Pretzels 4.3/30*100=14.3%

Popcorn 3.8/30*100=12.7%

Snack nuts 2.5/30*100=8.3%

A time series graph represents data that occur over a specific period of time.

The data show the percentage of U.S. adults who smoke. Draw and analyze a time series graph for the data.

20102000199019801970Year

1923253337Percent

Solution:

At an outpatient testing center, the number of cardiograms performed each day for 20 days is shown. Construct a stem and leaf plot for the data.

• The stem and leaf plot is a method of organizing data and is a combination of sorting and graphing.

• A stem and leaf plot is a data plot that uses part of the data value as the stem and part of the data value as the leaf to form groups or classes.

25 31 20 32 13

14 43 02 57 23

36 32 33 32 44

32 52 44 51 45

Solution:

Quantitative Data

• Histogram

• Frequency Polygon

• Ogive

Qualitative Data

• Bar Graph

• Pareto Chart

• Pie Graph

Time

• Time Series Graph

1. Cut off (or truncate) the y axis.

2. Exaggerate a one-dimensional increase by showing it in two dimensions.

3. Omit labels or units on the axes of the graph.

4. All graphs should contain a source for the information presented.