Dr. Fowler AFM Unit 8-2 Measures of Central Tendency Compute the mean, median, and mode of...

Dr. Fowler AFM Unit 8-2

Measures of Central Tendency

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

• Find the five-number summary of a distribution with the Box and Whisker Plot.

Section 15.2, Slide 3

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

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Section 15.2, Slide 4

• Example: Listed are the yearly earnings of some celebrities.

The Mean:

(continued on next slide)

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

b) Is this mean anaccurate measure of the “average” earnings for these celebrities?

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Section 15.2, Slide 5

• Solution (a): Summing the salaries and dividing by 10 gives us

The Mean:

Solution (b): 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.

Section 15.2, Slide 6

What is the Median?

1) Arrange the numbers in ascending order (smallest to greatest).

2) The median is the middle number. If there are 2 middle numbers, then the median is the average of the 2 middle numbers.

What is the Median of the celebrity salaries?

1)Arrange in ascending order:20, 30, 32, 33, 33, 40, 45, 45, 110, 260

2) The median is the middle number. Since there are 2 middle numbers, we add them and divide by 2:33+40 = 73divide by 2 = 36.5 million

Section 15.2, Slide 8

The Mode:

Section 15.2, Slide 9

c) no mode.

a) The mode is 5.a) 5, 5, 68, 69, 70

• Example: Find the mode for each data set.

The Mode:

b) 3, 3, 3, 2, 1, 4, 4, 9, 9, 9

c) 98, 99, 100, 101, 102

d) 2, 3, 4, 2, 3, 4, 5

b) 2 modes: 3 and 9.

d) no mode.

Section 15.2, Slide 10

The Five Number Summary

Section 15.2, Slide 11

• Example: Consider the list of ages of some of the past U.S. Presidents at inauguration:

42, 43, 46, 51, 51, 51, 52, 54, 55, 55, 56, 56, 60, 61, 61, 64, 69.

The Five Number Summary

(continued on next slide)

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

Section 15.2, Slide 12

• Solution:

The Five Number Summary

(continued on next slide)

(a): Finding the median, we can identify the lower and upper halves.

(b): The median of the lower half is

The median of the upper half is

Section 15.2, Slide 13

(c): The five number summary is We represent the five-number summary by a graph called a box-and-whisker plot.

The Five Number Summary

(continued on next slide)

Section 15.2, Slide 14

Box & Whisker on TI-84https://www.youtube.com/watch?v=iobQ-ge26kU

Excellent Job !!!Well Done

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© 2010 Pearson Education, Inc. All rights reserved. Section 15.2, Slide 17

The Mean and the Median

Section 15.2, Slide 18

• 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:

Section 15.2, Slide 19

• 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:

Section 15.2, Slide 20

• 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?

The Mean and the Median

Section 15.2, Slide 21

• 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 Mean and the Median

Section 15.2, Slide 22

• 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?

The Mean:

(continued on next slide)

Section 15.2, Slide 23

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

The mean is:

Section 15.2, Slide 24

• Example: Assume that you are negotiating the contract for your union. You have gathered annual wage data and found that three workers earn $30,000, five workers earn $32,000, three workers earn $44,000, and one worker earns $50,000. In your negotiations, which measure of central tendency should you emphasize?

Comparing Measures of Central Tendency

© 2010 Pearson Education, Inc. All rights reserved. Section 15.2, Slide 25

• Solution:Mode: $32,000

Median: $32,000

Mean:

The mean is $36,000.

To make the salaries appear as low as possible, you would want to use the mode and median.

Comparing Measures of Central Tendency