Chapter 2: Frequency Distributions and Graphs

Diana Pell

Section 2.1: Organizing Data

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

Categorical Frequency Distributions

Exercise 1. Twenty-five army inductees were given a blood test to determine their blood type.The data set is

Construct a frequency distribution for the data.

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

Grouped Frequency Distributions

Exercise 2. Suppose a researcher wished to do a study on the ages of the 50 wealthiest peoplein the world.

lower class limits:upper class limits:

Note: The class limits should have the same decimal place value as the data, but the classboundaries should have one additional place value and end in a 5.

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The class width for a class in a frequency distribution is found by subtracting the lower (orupper) class limit of one class from the lower (or upper) class limit of the next class.

Note: Do not subtract the limits of a single class. It will result in an incorrect answer.

A frequency distribution should abide the following rules:

1. There should be between 5 and 20 classes.

2. It is preferable but not absolutely necessary that the class width be an odd number. Theclass midpoint Xm is obtained by adding the lower and upper boundaries and dividing by2, or adding the lower and upper limits and dividing by 2

3. The classes must be mutually exclusive.

4. The classes must be continuous (no gaps).

5. The classes must be exhaustive. There should be enough classes to accommodate all the data.

6. The classes must be equal in width.

Constructing a Grouped Frequency Distribution

1. Determine the classes.

2. Find the highest and lowest values.

3. Find the range.

4. Select the number of classes desired.

5. Find the width by dividing the range by the number of classes and rounding up.

6. Select a starting point (usually the lowest value or any convenient number less than the lowestvalue); add the width to get the lower limits.

7. Find the upper class limits.

8. Find the boundaries.

9. Tally the data.

10. Find the numerical frequencies from the tallies, and find the cumulative frequencies.

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Exercise 3. These data represent the record high temperatures in degrees Fahrenheit (◦F ) foreach of the 50 states. Construct a grouped frequency distribution for the data, using 7 classes.

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Constructing an Ungrouped Frequency Distribution

Exercise 4. The data shown here represent the number of miles per gallon (mpg) that 30 se-lected four-wheel-drive sport utility vehicles obtained in city driving. Construct a frequencydistribution, and analyze the distribution.

The cumulative frequencies are:

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Exercise 5. The data represent the ages of our Presidents at the time they were first inaugu-rated.

a. Were the data obtained from a population or a sample? Explain your answer.

b. What was the age of the oldest President?

c. What was the age of the youngest President?

d. Construct a frequency distribution for the data using 7 classes.

e. Are there any peaks in the distribution?

f. Identify any possible outliers.

g. Write a brief summary of the nature of the data as shown in the frequency distribution.

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Section 2.2: Histograms, Frequency Polygons, and Ogives

The three most commonly used graphs in research are

1. The histogram.

2. The frequency polygon.

3. The cumulative frequency graph, or ogive.

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

Exercise 6. Construct a histogram to represent the data shown for the record high tempera-tures for each of the 50 states.

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

Exercise 7. Using the frequency distribution given in Exercise 6, construct a frequency polygon.

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The ogive is a graph that represents the cumulative frequencies for the classes in a frequencydistribution.

Exercise 8. Construct an ogive for the frequency distribution described in Exercise 4 above.

Distribution Shapes

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Exercise 9. Assume you are a realtor in Bradenton, Florida. You have recently obtained alisting of the selling prices of the homes that have sold in that area in the last 6 months. Youwish to organize those data so you will be able to provide potential buyers with useful informa-tion. Use the following data to create a histogram, frequency polygon, and cumulative frequencypolygon.

a. What questions could be answered more easily by looking at the histogram rather than thelisting of home prices?

b. What different questions could be answered more easily by looking at the frequency polygonrather than the listing of home prices?

c. What different questions could be answered more easily by looking at the cumulative frequencypolygon rather than the listing of home prices?

d. Are there any extremely large or extremely small data values compared to the other datavalues?

e. Which graph displays these extremes the best?

f. Is the distribution skewed?

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Section 2.3: Other Types of Graphs

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

Exercise 10. The table shows the average money spent by first-year college students. Draw ahorizontal and vertical bar graph for the data.

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Bar graphs can also be used to compare data for two or more groups. These types of bar graphsare called compound bar graphs. Consider the following data for the number (in millions) ofnever married adults in the United States.

Note: Read about a Pareto chart and time series graph!!!

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

Exercise 11. This frequency distribution shows the number of pounds of each snack food eatenduring the Super Bowl. Construct a pie graph for the data.

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A dotplot is a statistical graph in which each data value is plotted as a point (dot) above thehorizontal axis.

Exercise 12. The data show the number of named storms each year for the last 40 years.Construct and analyze a dotplot for the data.

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

Exercise 13. At an outpatient testing center, the number of cardiograms performed each dayfor 20 days is shown. Construct a stem and leaf plot for the data.

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Misleading Graphs

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