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Describing Data: Frequency Tables, Frequency Distributions, and Graphic Presentation

Chapter 2

Copyright ©2021 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

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This chapter uses data from the Applewood Auto Group to demonstrate how to create frequency tables and frequency distributions. Then the grouped data is presented graphically.

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Learning Objectives

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LO2-1 Summarize qualitative variables with frequency and relative frequency tables

LO2-2 Display a frequency table using a bar or pie chart

LO2-3 Summarize quantitative variables with frequency and relative frequency distributions

LO2-4 Display a frequency distribution using a histogram or frequency polygon

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Constructing Frequency Tables

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Mutually exclusive means the data fit in just one class

Collectively exhaustive means there is a class for each value

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FREQUENCY TABLE A grouping of qualitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

Frequency tables are used in descriptive statistics to show the pattern of the data, identify where values tend to concentrate, and expose extreme or unusual values. Using the Applewood Auto Group data, the classes are each of the four dealerships.

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Constructing Frequency Tables (2 of 3)

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To construct a frequency table

First sort the data into classes

Count the number in each class and report as the class frequency

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Microsoft Excel can create frequency tables with a tool called the Pivot Table. Instructions are given in Appendix C.

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Constructing Frequency Tables (3 of 3)

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Convert each frequency to a relative frequency

Each of the class frequencies is divided by the total number of observations

Shows the fraction of the total number observations in each class

The relative frequency captures the relationship between a class frequency and the total number of observations. Microsoft Excel can create frequency tables with a tool called the Pivot Table. Instructions are given in Appendix C.

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Graphic Presentation of Qualitative Data

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Use a bar chart when you wish to compare the number of observations for each class of a qualitative variable.

BAR CHART A graph that shows the qualitative classes on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are proportional to the heights of the bars.

Use a bar chart to graphically show the class frequencies. Excel software allows us to easily create bar charts. The instructions are in Appendix C.

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Graphic Presentation of Qualitative Data (2 of 2)

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Use a pie chart when you wish to compare relative differences in the percentage of observations for each class of a qualitative variable.

PIE CHART A chart that shows the proportion or percentage that each class represents of the total number of frequencies.

Use a pie chart to graphically show the relative frequencies. Excel software also easily creates pie charts and the instructions can be found in Appendix C.

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Constructing Frequency Distributions

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This is a four-step process

Decide on the number of classes

Determine the class interval

Set the individual class limits

Tally the data into classes and determine the number of the observations in each class

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FREQUENCY DISTRIBUTION A grouping of quantitative data into mutually exclusive and collectively exhaustive classes showing the number of observations in each class.

Earlier we used frequency tables for qualitative data. Now we learn how to present quantitative data; one technique for doing so is using frequency distributions.

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Frequency Distributions

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Step 1 Decide on the number of classes

Use the 2k > n rule, where n=180

k is the number of classes

n is the number of values in the data set

2k > 180, let k = 8

So use 8 classes

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Examining the raw data in the table, we find there are 180 values, so n = 180. First try k=7, 2 to the 7th power only equals 128 and 128 is not greater than 180 so 7 classes will not meet the criteria. Then try k=8, 2 to the eighth power equals 256 so decide to use 8 classes since 256 > 180.

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Frequency Distributions (2 of 4)

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Step 2 Determine the class interval, i

Round up to some convenient number

So decide to use an interval of $400

The interval is also referred to as the class width

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After analyzing the table of raw data, you find the maximum value is $3,292 and the minimum value is $294, and divide the difference by the number of classes. In this example, there are 8 classes so the result is $374.75. Round the result up to some convenient number like a multiple of 10 or 100; here we’ll use an interval of $400.

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Frequency Distributions (3 of 4)

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Step 3 Set the individual class limits

Lower limits should be rounded to an easy-to-read number when possible

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Make sure to use the words “up to” in each class in order to construct mutually exclusive classes; this will ensure each value in the data set fits in one and only one class. Also, check to see that the minimum value will go in the first class and the maximum value will go in the last class when setting class limits. Here the minimum value of $294 goes in the first class and the maximum value of $3,292 will go in the last class. This ensures you have exhaustive classes. Class midpoints can represent a typical value for each class and can be determined by adding the lower limits of consecutive classes and dividing by 2.

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Frequency Distributions (4 of 4)

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Step 4 Tally the individual data into the classes and determine the number of observations in each class

The number of observations is the class frequency

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There is some loss of detail when data is grouped, but the frequency distribution results in an understandable and organized form. Equal class widths are preferred but might not always be possible.

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Relative Frequency Distributions

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To find the relative frequencies, simply take the class frequency and divide by the total number of observations

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The relative class frequencies shows each class frequency relative to the entire distribution and adds up to 1.000.

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Graphic Presentation of a Frequency Distribution

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A histogram shows the shape of a distribution

Each class is depicted as a rectangle, with the height of the bar representing the number in each class

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HISTOGRAM A graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars, and the bars are drawn adjacent to each other.

Histograms allow us to get a quick picture of the main characteristics of the data. They are similar to bar charts, but since this is continuous data, there are no gaps between the bars. They are drawn adjacent to one another.

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Graphical Presentation of a Frequency Distribution

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A frequency polygon, similar to a histogram, also shows the shape of a distribution

These are good to use when comparing two or more distributions

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A frequency polygon consists of line segments connecting the points formed by the intersection of the class midpoints and the class frequencies. You may recall that class midpoints are halfway between the lower limits of two consecutive classes and represent the typical value for each class. Since the definition of a polygon is that it is a closed plane, the frequency polygon is closed by anchoring the connected line segments to the X axis at zero frequency one interval higher than the highest midpoint and one interval lower than the lowest midpoint.

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Cumulative Frequency Distributions

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To construct a cumulative frequency distribution, add each frequency to the frequencies before it

This shows how many values have accumulated as you move from one class down to the next class

To construct a cumulative frequency distribution, refer to the preceding table and note that there were eight vehicles in which the profit earned was less than $600. Those 8 vehicles, plus the 11 in the next higher class, for a total of 19, earned a profit of less than $1,000. The cumulative frequency for the next higher class is 42, found by 8 + 11 + 23. This process is continued for all the classes. All the vehicles earned a profit of less than $3,400.

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Cumulative Relative Frequency Distribution

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To construct a cumulative relative frequency distribution, we divide the cumulative frequencies by the total number of observations

As shown in Table 2-9, the cumulative relative frequency of the fourth class is 80/180 = 44%. This means that 44% of the vehicles sold for less than $1,800.

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Cumulative Frequency Polygon

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To plot a cumulative frequency distribution, scale the upper limit of each class along the X-axis and the corresponding cumulative frequencies along the Y-axis. Label the vertical axis on the right in terms of cumulative relative frequencies.

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Chapter 2 Practice Problems

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Question 5

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Wellstone Inc. produces and markets replacement covers for cell phones in five different colors: bright white, metallic black, magnetic lime, tangerine orange, and fusion red. To estimate the demand for each color, the company set up a kiosk for several hours in the Mall of America and asked randomly selected people which cover color was their favorite.

What is the table called?

Draw a bar chart for the table.

Draw a pie chart.

If Wellstone Inc. plans to produce one million cell phone covers, how many of each color should it produce?

LO2-2

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Question 11

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Wachesaw Manufacturing Inc. produced the following number of units in the last 16 days.

The information is to be organized into a frequency distribution.

How many classes would you recommend?

What class interval would you suggest?

What lower limit would you recommend for the first class?

Organize the information into a frequency distribution and determine the relative frequency distribution.

Comment on the shape of the distribution.

LO2-3

Question 17

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The following frequency distribution reports the number of frequent flier miles, reported in thousands, for employees of Brumley Statistical Consulting Inc. during the most recent quarter.

How many employees were studied?

What is the midpoint of the first class? What lower limit would you recommend for the first class?

Construct a histogram.

A frequency polygon is to be drawn. What are the coordinates of the plot for the first class?

Construct a frequency polygon.

Interpret the frequent flier miles accumulated using the two charts.

LO2-4