15.2 Measures of Central TendencyObjectives:

• Compute the mean, median, and mode of distributions.

• Find the five-number summary of a distribution.

• Apply measures of central tendency to compare data sets.

The Mean

• We use the Greek letter Σ (capital sigma) to indicate a sum. For example, we will write the sum of the data values 7, 2, 9, 4, and 10 by Σx = 7 + 2 + 9 + 4 + 10.

• We represent the mean of a sample of a population by (read as “x bar”), and we will use the Greek letter μ (lowercase mu) to represent the mean of the whole population.

x

The Mean cont. (2)

Example: A car company has been studying its safety record at a factory and found that the number of accidents over the past 5 years was 25, 23, 27, 22, and 26. Find the mean annual number of accidents for this 5-year period.

Solution: We add the number of accidents and divide by 5.

The Mean cont. (3)

Example: The water temperature at a point downstream from a plant for the last 30 days is summarized in the table. What is the mean temperature for this distribution?

Solution: A third column is added to the table that contains the products of the raw scores and their frequencies.

The Mean cont. (3)

The mean is

The Mean and the Median Example: Listed are the yearly earnings of some celebrities.

a) What is the mean of the earnings of the celebrities on this list?

b) Is this mean an accurate measure of the “average” earnings for these celebrities?Eight of the celebrities have earnings below the mean, whereas only two have earnings above the mean. The mean in this example does not give an accurate sense of what is “average” in this set of data because it was unduly influenced by higher earnings.

The Mean and the Median cont. (2)

a) What is the median of the earnings of the celebrities on this list?

b) Is this median an accurate measure of the “average” earnings for these celebrities?

The Median

Example: The table lists the ages at inauguration of the presidents who assumed office between 1901 and 1993. Find the median age for this distribution.

Solution: We first arrange the ages in order to get

There are 17 ages. The middle age is the ninth, which is 55.

The Median cont. (2)Example: Fifty 32-ounce quarts of a particular brand of milk

were purchased and the actual volume determined. The results of this survey are reported in the table. What is the median for this distribution?

Solution: Because the 50 scores are in increasing order, the two middle scores are in positions 25 and 26. We see that 29 ounces is in position 25 and 30 ounces is in position 26. The median for this distribution is

The Five Number Summary

Example: Consider the list of ages of the presidents from a previous example: 42, 43, 46, 51, 51, 51, 52, 54, 55, 55, 56, 56, 60, 61, 61, 64, 69.

Find the following for this data set:a) the lower and upper halvesb) the first and third quartilesc) the five-number summary

The Five Number Summary cont. (2)Solution:a) Finding the median, we can identify the lower and upper

halves.

b) The First Quartile is the median of the lower half

The Third Quartile is the median of the upper half

c) The five number summary is

Box-and-whisker Plot

We represent the five-number summary by a graph called a box-and-whisker plot.

The Mode

Example: Find the mode for each data set.a) 5, 5, 68, 69, 70 b) 3, 3, 3, 2, 1, 4, 4, 9, 9, 9c) 98, 99, 100, 101, 102 d) 2, 3, 4, 2, 3, 4, 5Solution: a) The mode is 5.b) There are two modes: 3 and 9.

In (c) and (d), there is no mode.

Comparing Measures of Central Tendency

• Suppose you run a hat shop and have your customers hat sizes in a data base. You have found an excellent deal on a popular style but the deal is a loss leader for the supplier and they will only provide one size. Which measure of central tendency should you use?

• Assume that you are negotiating the contract for your union. You have gathered annual wage data and found that

nine workers earn $30,000 and one earns $140,000. In your negotiations, which measure of central tendency should you emphasize?

• In the above example, assume that you are negotiating the contract for management. Which measure of central tendency would you emphasize?