# Measures of Central Tendency

1. Each of the three measures of central tendency?the mean, the median, and the mode?are more appropriate for certain populations than others.

Search the Cybrary and/or Internet. For each type of measure, give two examples of populations where it would be the most appropriate indication of central tendency.

2. Find the mean, median, and mode of the following data set:

5 15 9 22 67 42 2 72 81 53 6 70 41 9 42 23

3. Sometimes, we can take a weighted approach to calculating the mean. Take our example of high temperatures in July. Suppose it was 98°F on 7 days, 96°F on 14 days, 88°F on 1 day, 100°F on 6 days and 102°F on 3 days. Rather than adding up 31 numbers, we can find the mean by doing the following:

Mean = ( 1 x 88 + 14 x 96 + 7 x 98 + 6 x 100 + 3 x 102) / 31

...where 1, 14, 7, 6, and 3 are the weights or frequency of a particular temperature's occurrence. Then we divide by the total of number of occurrences.

Suppose we are tracking the number of home runs hit by the Boston Red Sox during the month of August:

Number of Games HRs Hit each Day

2 3

5 2

6 1

7 0

Using the weighted approach, calculate the average number of home runs per game hit by the Sox.

4. When a pair of dice is rolled, the total will range from 2 (1,1) to 12 (6,6). It is a fact that some numbers will occur more frequently than others as the dice are rolled over and over.

Why will some numbers come up more frequently than others?

Each die has six sides numbered from 1 to 6. How many possible ways can a number be rolled? In other words, we can roll (2,3) or (3,2) or (6,1) and so on. What are the total (x,y) outcomes that can occur?

How might you then estimate the percentage of the time a particular number will come up if the dice are rolled over and over?

Once these percentages have been calculated, how might the mean value of the all the numbers thrown be determined?

If you have completed the Discussion Board assignment, you have an idea of what a population distribution is.

There is a very famous distribution that describes the frequency of the number of times a number comes up in a series of dice rolls. Use the Library or the Internet to see if you can find its name.

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#### Solution Preview

1. Each of the three measures of central tendency?the mean, the median, and the mode?are more appropriate for certain populations than others.

Search the Cybrary and/or Internet. For each type of measure, give two examples of populations where it would be the most appropriate indication of central tendency.

The mean (average) of a population is most appropriate when the data are normal or symmetrically distributed, and there are no outliers. When there are outliers, the mean might be inappropriately high or low. You could use the mean when measuring things like height or weight, as long as there aren't any outliers.

The median (middle value) of a population is most appropriate when the data is not symmetrical or there are outliers. Outliers don't affect the value of the median as much as they affect the value of the mean. Often, you use the median for things like income or temperature, which tend to have outlying values.

The mode (the number that occurs the most frequently) is not often used in statistics. It would be appropriate in a situation such as asking consumers which product they liked the best - you wouldn't be able to average the results (out of products 1, 2, and 3, there is no product 1.5, but it would be useful to say that ...

#### Solution Summary

The solution consists of complete answers and explanations to four statistics questions. These questions are on measures of central tendency and a specific kind of population distribution.