A partial relative frequency distribution is given

Tabular and Graphical Presentations Statistics (exercises)

Aleksandra Pawłowska

April 7, 2020

Glossary (part 1)

Categorical data Labels or names used to identify categories of like items. Quantitative data Numerical values that indicate how much or how many. Frequency distribution A tabular summary of data showing the number (fre- quency) of data values in each of several nonoverlapping classes. Relative frequency distribution A tabular summary of data showing the fraction or proportion of data values in each of several nonoverlapping classes. Percent frequency distribution A tabular summary of data showing the percent- age of data values in each of several nonoverlapping classes. Bar chart A graphical device for depicting qualitative data that have been sum- marized in a frequency, relative frequency, or percent frequency distribution. Pie chart A graphical device for presenting data summaries based on subdivision of a circle into sectors that correspond to the relative frequency for each class. Dot plot A graphical device that summarizes data by the number of dots above each data value on the horizontal axis.

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Glossary (part 2)

Histogram A graphical presentation of a frequency distribution, relative frequency distribution, or percent frequency distribution of quantitative data constructed by placing the class intervals on the horizontal axis and the frequencies, relative frequencies, or percent frequencies on the vertical axis. Cumulative frequency distribution A tabular summary of quantitative data show- ing the number of data values that are less than or equal to the upper class limit of each class. Cumulative relative frequency distribution A tabular summary of quantitative data showing the fraction or proportion of data values that are less than or equal to the upper class limit of each class. Cumulative percent frequency distribution A tabular summary of quantitative data showing the percentage of data values that are less than or equal to the upper class limit of each class. Ogive A graph of a cumulative distribution. Scatter diagram A graphical presentation of the relationship between two quan- titative variables. One variable is shown on the horizontal axis and the other variable is shown on the vertical axis. Trendline A line that provides an approximation of the relationship between two variables.

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Useful tips (part 1)

1 Often the number of classes in a frequency distribution is the same as the number of categories found in the data. Most statisticians recommend that classes with smaller frequencies be grouped into an aggregate class called „other”. Classes with frequencies of 5% or less would most often be treated in this fashion.

2 The sum of the frequencies in any frequency distribution always equals the number of observations. The sum of the relative frequencies in any relative frequency distribution always equals 1.00, and the sum of the percentages in a percent frequency distribution always equals 100.

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Useful tips (part 2)

1 A bar chart and a histogram are essentially the same thing; both are graphical presentations of the data in a frequency distribution. A histogram is just a bar chart with no separation between bars. For some discrete quantitative data, a separation between bars is also appropriate.

2 The last entry in a cumulative frequency distribution always equals the total number of observations. The last entry in a cumulative relative frequency distribution always equals 1.00 and the last entry in a cumulative percent frequency distribution always equals 100.

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Exercises

Task 1

The response to a question has three alternatives: A, B, and C. A sample of 120 responses provides 60 A, 24 B, and 36 C. Show the frequency and relative frequency distributions.

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Task 1 – solution

frequency distribution relative frequency distribution A 60 0.5 B 24 0.2 C 36 0.3 Total 120 1.00

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Task 2

A questionnaire provides 58 Yes, 42 No, and 20 no-opinion answers.

1 In the construction of a pie chart, how many degrees would be in the section of the pie showing the Yes answers?

2 How many degrees would be in the section of the pie showing the No answers?

3 Construct a pie chart. 4 Construct a bar chart.

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Task 2 – solution 1 In the construction of a pie chart, how many degrees would be in the

section of the pie showing the Yes answers? 174 2 How many degrees would be in the section of the pie showing the No

answers? 126 3 Construct a pie chart.

4 Construct a bar chart.

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Task 3

Consider the following frequency distribution.

Construct a cumulative frequency distribution and a cumulative relative frequency distribution.

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Task 3 – solution

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Task 4

A doctor’s office staff studied the waiting times for patients who ar- rive at the office with a request for emergency service. The following data with waiting times in minutes were collected over a one-month period.

Use classes of 0–4, 5–9, and so on in the following:

1 Show the frequency distribution. 2 Show the relative frequency distribution. 3 Show the cumulative frequency distribution. 4 Show the cumulative relative frequency distribution. 5 What proportion of patients needing emergency service wait 9

minutes or less?

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Task 4 – solution

1 Show the frequency distribution. Show the relative frequency distribution.

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Task 4 – solution

1 Show the cumulative frequency distribution. Show the cumulative relative frequency distribution.

2 What proportion of patients needing emergency service wait 9 minutes or less? 0.6

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Task 5

The following 20 observations are for two quantitative variables, x and y .

1 Develop a scatter diagram for the relationship between x and y .

2 What is the relationship, if any, between x and y?

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Task 5 – solution

A negative relationship between x and y ; y decreases as x increases.

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Task 6

A partial relative frequency distribution is given.

1 What is the relative frequency of class D? 2 The total sample size is 200. What is the frequency of class

D? 3 Show the frequency distribution. 4 Show the percent frequency distribution.

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Task 6 – solution

1 What is the relative frequency of class D? 0.20 2 The total sample size is 200. What is the frequency of class

D? 40 3 Show the frequency distribution. Show the percent frequency

distribution.

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Task 7

Data for a sample of 55 members of the Baseball Hall of Fame in Cooperstown, New York, are shown on next slide. Each observation indicates the primary position played by the Hall of Famers: pitcher (P), catcher (H), 1st base (1), 2nd base (2), 3rd base (3), shortstop (S), left field (L), center field (C), and right field (R).

1 Use frequency and relative frequency distributions to summarize the data.

2 What position provides the most Hall of Famers? 3 What position provides the fewest Hall of Famers? 4 What outfield position (L, C, or R) provides the most Hall of

Famers? 5 Compare infielders (1, 2, 3, and S) to outfielders (L, C, and

R).

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Task 7 – solution

1 Use frequency and relative frequency distributions to summarize the data.

2 What position provides the most Hall of Famers? Pitcher

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Task 7 – solution

1 What position provides the fewest Hall of Famers? 3rd base 2 What outfield position (L, C, or R) provides the most Hall of

Famers? Right field 3 Compare infielders (1, 2, 3, and S) to outfielders (L, C, and

R). Infielders 16 to outfielders 18

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