# Measures of Tendency and Measures of Variation

Find the mean for each data set. Round to the nearest tenth.

1. Starting teaching salaries (US Dollars):

$38,400, $39,720, $28, 458, $29,679, $33, 679

2. Find the mean of each distribution. Round to the nearest tenth.

Scores on a quiz, on a scale from 0 to 10:

Value Frequency

7 4

8 6

9 7

10 11

3. Find the mode or modes for each of the given list of numbers.

Ages (years) of children in a day-care facility.

1,2,2,1,2,2,1,1,2,2,3,4,2,3,4,2,3,2,3

3. The following table gives the value (in millions of dollars) of the 10 most values baseball teams as estimated by Forbes in 2007.

Rank Team Value

1 New York Yankees 1306

2 New York Mets 824

3 Boston Red Sox 816

4 Los Angeles Dodgers 694

5 Chicago Cubs 642

6 Los Angeles Angels of Anaheim 500

7 Atlanta Braves 497

8 San Francisco Giants 494

9 St. Louis Cardinals 484

10 Philadelphia Phillies 481

- Find the mean value of these teams.

- Find the median value of these teams.

- What might account for the difference between these values?

Measures of Variation

4. Expenditures (in millions of dollars) for various government services in 2005 are given are given for the five largest counties in the United States by population:

Los Angeles, CA; Cook, IL; Harris, TX; Maricopa, AZ; and Orange, CA.

Find the range and the standard deviation for each given category.

Parks and Recreation: 227, 112, 26, 7, 1

5. Find the standard deviation for the grouped data:

Scores on a calculus exam:

Scores Frequency

30-39 1

40-49 6

50-59 13

60-69 22

70-79 17

80-89 13

90-99 8

6. An assembly-line machine turns out washers with the following thickness (in millimeters)

Find the mean and standard deviation of these thicknesses.

1.20 1.01 1.25 2.20 2.58 2.19 1.29 1.15

2.05 1.46 1.90 2.03 2.13 1.86 1.65 2.27

1.64 2.19 2.25 2.08 1.96 1.83 1.17 2.24

7. The Quaker Oats Company conducted a survey to determine whether a proposed premium, to be included with purchases of the firm's cereal, was appealing enough to generate new sales. Four cities were used as test markets, where the cereal was distributed with the premium, and four cities were used as controlled markets, where the cereal was distributed without the premium. The eight cities were chosen on the basis of their similarity in terms of population, per-capita income, and total cereal purchase volume. The results were as follows:

Please see the attached document for the table.

- Find the mean of the change in market share for the four cities.

- Find the mean of the change in market share for the four control cities.

- Find the standard deviation of the change in market share for the test cities.

- Find the standard deviation of the change in market share in control cities.

- Find the difference between the mean of part (a) and the mean of part (b). This represents the estimate of the percent change in sales due to the premium.

The two standard deviations from part (c) and part (d) were used to calculate an "error" of ± 7.95 for the estimate in part - With the amount of error, what is the smallest and largest estimate of the increase in sales?

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#### Solution Preview

Measures of Central Tendency

Section 10.2

Find the mean for each data set. Round to the nearest tenth.

2. Starting teaching salaries (US Dollars):

$38,400, $39,720, $28, 458, $29,679, $33, 679

The mean value of a dataset (represented symbolically by ) is calculated by adding up all of the values in the dataset (represented symbolically by or sometimes by just ) and then dividing this sum by the total number of values in the dataset (represented symbolically by ). In this case you have: , exact to the tenths position (no rounding needed).

Find the mean of each distribution. Round to the nearest tenth.

6. Scores on a quiz, on a scale from 0 to 10:

Value Frequency

7 4

8 6

9 7

10 11

In the case of a frequency distribution like this one, if you first multiply each distinct value by its frequency and then add these products up you will have the sum of all of the values in the dataset. To calculate the mean then, all you need to do is divide the sum of the products by the sum of the frequency column (which is equal to the total number of values in the dataset). In this case you have:

, rounded off to the nearest tenth.

Find the mode or modes for each of the given list of numbers.

16. Ages (years) of children in a day-care facility.

1,2,2,1,2,2,1,1,2,2,3,4,2,3,4,2,3,2,3

The strict definition of the mode of a dataset is that the mode of a dataset is the value in the dataset that occurs most often, or more frequently than any other value in the dataset. In this dataset the value "1" occurs four times; the value "2" occurs nine times; the value "3" occurs four times, and; the value "4" occurs two times. Since the value "2" occurs more often than any other value in this dataset, the mode for this dataset is the value "2".

26. The following table gives the value (in millions of dollars) of the 10 most values baseball teams as estimated by Forbes in 2007.

Rank Team Value

1 New York Yankees 1306

2 New York Mets 824

3 Boston Red Sox 816

4 Los Angeles Dodgers 694

5 Chicago Cubs 642

6 Los Angeles Angels of Anaheim 500

7 Atlanta Braves 497

8 San Francisco Giants 494

9 St. Louis Cardinals 484

10 Philadelphia Phillies 481

a. Find the mean value of these teams.

millions of dollars

b. Find the median value of these teams.

To find the median the first thing you must do is rank-order the values in the dataset from the smallest to the largest, which has already been done for us in the table. Next find THE value that sits in the middle of the rankings ...

#### Solution Summary

The measures of tendency and measures of variations are examined.

Measures of Central Tendency and Variation and their Interpretation

A quality characteristic of interest for a tea-bag-filling process is the weight of the tea in the individual bags. If the bags are under filled, two problems arise. First, customers may not be able to brew the tea to be as strong as they wish. Second, the company may be in violation of the truth-in-labeling laws. In this example, the label weight on the package indicates that, on average, there are 3.40 grams of tea in a bag. If the average amount of tea in a bag exceeds the label weight, the company is giving away product. Getting an exact amount of tea in a bag is problematic because of variation in the temperature and humidity inside the factory, differences in the density of the tea, and the extremely fast filling operation of the machine (approximately 215 bags a minute). The attached table provides the weight in grams of a sample of 50 bags produced in one hour by a single machine.

a. Compute the arithmetic mean and median.

b. Compute the first quartile and third quartile.

c. Compute the range, interquartile range, variance, standard deviation, and coefficient of variation.

d. Interpret the measures of central tendency within the context of this problem. Why should the company producing the tea bags be concerned about the central tendency?

e. Interpret the measures of variation within the context of this problem. Why should the company producing the tea bags be concerned about variation?