07 Jan Given the following frequency

MATH399 Applied Managerial Statistics

Week 1 Assignment Using Measures of Central Tendency
Question

Given the following frequency table of values, is the mean or the median likely to be a better measure of the center of the data set?

Value323334353637Frequency236221

QuestionGiven the following list of values, is the mean or the median likely to be a better measure of the center of the data set?

25, 29, 23, 26, 25, 27, 10, 26, 23, 23, 26

QuestionGiven the following list of values, is the mean or the median likely to be a better measure of the center of the data set?

26, 27, 26, 30, 27, 31, 29, 28, 53, 29

Determining the Best Measure of Center

Key Terms

• Mean: the sum of all the items in a list divided by the number of items in the list

The term Mean is often used interchangeably with the term Average

• Median: a number that splits a data set in half, with one half smaller and one half larger; the center or middle value of a data set

• Mode: the number(s) that occurs most often in a data set

• Outliers: values that are very different from the rest of the values in a data set

QuestionGiven the following frequency table of values, is the mean or the median likely to be a better measure of the center of the data set?

Value252627282930Frequency223362

QuestionGiven the following frequency table of values, is the mean or the median likely to be a better measure of the center of the data set?

Value27282930313233343536373839404142434445Frequency1000000000001215223

QuestionGiven the following list of values, is the mean or the median likely to be a better measure of the center of the data set?

35, 39, 37, 37, 38, 34, 35, 37, 65

QuestionGiven the following list of values, is the mean or the median likely to be a better measure of the center of the data set?

29, 30, 31, 30, 30, 34, 32, 28

Determining the Best Measure of Center

With three different measures of central tendency—mean, median, and mode—it is sometimes difficult to decide which is the best option to summarize your data. Let’s look at each measure and discuss when we should use it.

Mode

The mode is most often used with nominal (categorical) and ordinal data, when it is not meaningful to take the mean or the median. For example, to summarize a survey of voters’ preferred candidate, it would make sense to summarize the data by saying which candidate was most preferred. That would be the mode in this case (the value which occurs the most often).

Similarly, to summarize customer satisfaction on a survey, where the choices might be “Very Dissatisfied”, “Dissatisfied”, “Satisfied”, or “Very Satisfied” it would be reasonable to identify the category which most customers chose, rather than trying to use the mean or median.

Mean and Median

The mean is useful in data sets that do not have any outliers, which are values that are very different from the rest of the values. The reason that outliers are bad for the mean is that a few very large (or very small) values can pull the mean up or down, which in many cases distorts or misrepresents the data.

If a data set has outliers, then it is typically better to use the median, since it will not be distorted by a few very large or very small values.

The histogram below shows the frequencies of a data set. As you can see, there are many small values with high frequencies, and a few outliers which are much larger. The mean and the median are also labeled with dotted lines. As you can see, the mean has been pulled up by the large outliers, whereas the median gives a better idea of the middle of the data.

A coordinate plane has a horizontal axis titled Value labeled from 0 to 16 in increments of 2 and a vertical axis titled Frequency labeled from 0 to 8 in increments of 2. Bars are plotted on the graph where the horizontal coordinate value is listed first and the vertical coordinate value is listed second: 1,7; 2, 8; 3, 6; 4, 2; 5, 2; 6, 1; 10, 1; 11, 1; 15, 1. A vertical dashed line at 2 on the Value axis is labeled Median. A vertical dashed line at approximately 3.5 on the Value axis is labeled Mean.

Key Terms

• Mean: the sum of all the items in a list divided by the number of items in the list

The term Mean is often used interchangeably with the term Average

• Median: a number that splits a data set in half, with one half smaller and one half larger; the center or middle value of a data set

• Mode: the number(s) that occurs most often in a data set

• Outliers: values that are very different from the rest of the values in a data set

In the real world, household income often has a distribution like the above one, where most household incomes are low or moderate, but there are also a few very large values. In such situations, the median income is often a better representative of the typical household than the mean.

Example

QuestionWould the mean or the median be a better measure of central tendency for the following list of data?

6, 6, 7, 8, 9, 30

QuestionGiven the following frequency table of values, is the mean or the median likely to be a better measure of the center of the data set?

Value2021222324Frequency73325

QuestionGiven the following list of values, is the mean or the median likely to be a better measure of the center of the data set?

31, 32, 31, 34, 30, 30, 31, 31, 30, 31

QuestionGiven the following frequency table of values, is the mean or the median likely to be a better measure of the center of the data set?

Value40414243444546474849505152535455565758596061626364656667686970Frequency5612330000000000000000000000001

QuestionGiven the following frequency table of values, is the mean or the median likely to be a better measure of the center of the data set?

Value3233343536373839404142434445464748495051525354555657Frequency32302400000000000000000001

QuestionGiven the following list of values, is the mean or the median likely to be a better measure of the center of the data set?

26, 27, 26, 30, 27, 31, 29, 28, 53, 29

Determining the Best Measure of Center

Key Terms

• Mean: the sum of all the items in a list divided by the number of items in the list

The term Mean is often used interchangeably with the term Average

• Median: a number that splits a data set in half, with one half smaller and one half larger; the center or middle value of a data set

• Mode: the number(s) that occurs most often in a data set

• Outliers: values that are very different from the rest of the values in a data set

QuestionGiven the following list of values, is the mean or the median likely to be a better measure of the center of the data set?

33, 33, 33, 31, 31, 34, 32, 31

QuestionGiven the following list of values, is the mean or the median likely to be a better measure of the center of the data set?

29, 56, 27, 29, 27, 28, 28, 30, 30, 27

QuestionEmily likes to catch and release fish in a pond and record their lengths. Estimate the mean of the lengths (in inches) of the fish given in the following grouped frequency table.

• Round the final answer to two decimal places.

Value Interval

Frequency

2-5

10

6-9

12

10-13

6

14-17

7

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QuestionThe frequency table below summarizes a list of recorded lengths (in inches) of randomly sampled fish in a pond. Find the mean.

Value

Frequency

5

3

6

1

7

1

8

2

9

3

10

4

11

6

12

1

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QuestionA surveyor would like to find the mean number of pets living in apartments in a city. He collects data from 36 apartments in the area. The graph shows the frequency for the number of pets living in the apartments.

Find the mean number of pets living in the 36 city apartments, and round your answer to the nearest tenth. Record your answer by dragging the purple point to the mean.

QuestionA student at Pine Valley high school would like to find the mean number of extra credit points her fellow students earned on a statistics chapter test. She collects data from 11 students in her class. The graph shows the frequency for the number of extra credit points earned by her fellow classmates.

Find the mean number of extra credit points earned by the 11 students, and round your answer to the nearest tenth. Record your answer by dragging the purple point to the mean.

QuestionFind the median of the numbers in the following list.

29,29,7,11,5,5

QuestionFind the mode of the following number of countries randomly selected travelers visited in the past two years.

13,13,15,7,8,4,4,7,13

QuestionFind the median of the following list of inches traveled by randomly selected worms in a two minute time period.

11,7,5,12,20,6

QuestionFind the mode of the following number of sweaters purchased by randomly selected customers at a department store.

12,11,5,12,4,5,12,3

QuestionA trainer would like to find the mean number of sports drinks the people in her class had in the last week. She collects data from 26 participants in her aerobics class. The graph shows the frequency for the number of sports drinks.

Find the mean number of sports drinks consumed by the 26 participants, and round your answer to the nearest tenth. Record your answer by dragging the purple point to the mean.

QuestionGiven the following list of tips (in dollars) earned in the last hour by waiters in a Japanese restaurant, find the median.

30,17,47,47,26,17,36,31,21,17

QuestionGiven the following frequency table of values, is the mean or the median likely to be a better measure of the center of the data set?

Value202122232425262728293031323334353637Frequency100000000000034331

QuestionGiven the following list of values, is the mean or the median likely to be a better measure of the center of the data set?

39, 41, 38, 39, 38, 41, 41, 39, 40

QuestionA student would like to find the mean number of people living in households in a neighborhood. He collects data from 37 homes in the area. The graph shows the frequency for the number of people living in the homes.

Find the mean number of people living in the 37 homes, and round your answer to the nearest tenth. Record your answer by dragging the purple point to the mean.

QuestionFind the median of the following list of dollars spent per customer at a cheese shop in the last hour.

32,19,21,16,27,15

QuestionEstimate the mean of the amounts (in dollars) randomly selected customers spent on chocolate chip cookies at a winter fair given in the following grouped frequency table.

• Round the final answer to one decimal place.

Value Interval

Frequency

0-3

5

4-7

6

8-11

13

12-15

1

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QuestionThe frequency table below summarizes a list of the amounts (in dollars) randomly selected customers spent on hot chocolate during a winter festival. Find the mean.

Value

Frequency

8

5

9

2

10

5

11

2

12

2

13

2

14

2

15

3

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QuestionFind the mode of the following amounts of exercise (in hours) randomly selected runners completed during a weekend.

2,14,14,4,2,4,1,14,4,4,8

QuestionFind the mode of the following list of points earned on a 16 point quiz given during a finance class.

7,7,3,2,7,16,12,16,12