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Frequency Distributions

A frequency distributionfrequency distribution is a table used to organize data. The left column (called classes or groups) includes numerical intervals on a variable being studied. The right column is a list of the frequencies, or number of observations, for each class. Intervals are normally of equal size, must cover the range of the sample observations, and be non-overlapping.

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Construction of a Frequency Distribution

Rule 1: Intervals (classes) must be inclusive and non-overlapping;

Rule 2: Determine k, the number of classes; Rule 3: Intervals should be the same width, w;

the width is determined by the following:

Both k and w should be rounded upward, possibly to the next largest integer.

Intervals ofNumber

Number)Smallest -Number (Largest Width Interval w

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Construction of a Frequency Distribution

Quick Guide to Number of Classes for a Frequency Distribution

Sample Size Number of Classes

Fewer than 50 5 – 6 classes

50 to 100 6 – 8 classes

over 100 8 – 10 classes

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Cumulative Frequency Distributions

A cumulative frequency distributioncumulative frequency distribution contains the number of observations whose values are less than the upper limit of each interval. It is constructed by adding the frequencies of all frequency distribution intervals up to and including the present interval.

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Relative Cumulative Frequency Distributions

A relative cumulative frequency relative cumulative frequency distributiondistribution converts all cumulative frequencies to cumulative percentages

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Histograms and Ogives

A histogramhistogram is a bar graph that consists of vertical bars constructed on a horizontal line that is marked off with intervals for the variable being displayed. The intervals correspond to those in a frequency distribution table. The height of each bar is proportional to the number of observations in that interval.

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Histograms and Ogives

An ogiveogive,, sometimes called a cumulative line graph, is a line that connects points that are the cumulative percentage of observations below the upper limit of each class in a cumulative frequency distribution.

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Histogram and Ogive for Example 2.1

Histogram of Weights for Example 2.1

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224.5 229.5 234.5 239.5 244.5 249.5

Interval Weights (mL)

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Stem-and-Leaf Display

A stem-and-leaf displaystem-and-leaf display is an exploratory data analysis graph that is an alternative to the histogram. Data are grouped according to their leading digits (called the stem) while listing the final digits (called leaves) separately for each member of a class. The leaves are displayed individually in ascending order after each of the stems.

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Stem-and-Leaf Display

Stem-and-Leaf Display

Stem unit: 10

9 1 1 2 4 6 7 8 8 9 9(9) 2 1 2 2 2 4 6 8 9 9

7 3 0 1 2 3 42 4 0 2

Stem-and-Leaf Display for Gilotti’s Deli Example

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Tables- Bar and Pie Charts -

IndustryNumber ofEmployees Percent

Tourism 85,287 0.35Retail 49,424 0.2Health Care 39,588 0.16Restaurants 16,050 0.06Communications 11,750 0.05Technology 11,144 0.05Space 11,418 0.05Other 21,336 0.08

Frequency and Relative Frequency Distribution for Top Company Employers Example

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Tables- Bar and Pie Charts -

1999 Top Company Employers in Central Florida

0.35

0.20.16

0.06 0.05 0.05 0.05 0.08

Touris

mReta

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Health C

are

Restaur

ants

Communic

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s

Techn

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Space

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Industry Category

Figure 2.9 Bar Chart for Top Company

Employers Example

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Tables- Bar and Pie Charts -

Figure 2.10 Pie Chart for Top Company Employers Example

1999 Top Company Employers in Central Florida

Tourism35%

Retail20%

Health Care16%

Others29%

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Pareto Diagrams

A Pareto diagramPareto diagram is a bar chart that displays the frequency of defect causes. The bar at the left indicates the most frequent cause and bars to the right indicate causes in decreasing frequency. A Pareto diagramPareto diagram is use to separate the “vital fewvital few” from the “trivial many.trivial many.”

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Line Charts

A line chartline chart, , also called a time plottime plot, , is a series of data plotted at various time intervals. Measuring time along the horizontal axis and the numerical quantity of interest along the vertical axis yields a point on the graph for each observation. Joining points adjacent in time by straight lines produces a time plot.

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Line Charts

Growth Trends in Internet Use by Age 1997 to 1999

16.520.2

26.331.3 32.7

9.813.8 15.8 17.2 18.5

5 7.511.4 13 14.2

05

101520253035

Apr-9

7

Jul-9

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Oct-97

Jan-

98

Apr-9

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Jul-9

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April 1997 to July 1999

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Age 18 to 29

Age 30 to 49

Age 50+