Measures of Central Tendency 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.

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.

The Mean: Example: Listed are the yearly earnings of some celebrities. What is the mean of the earnings of the celebrities on this list? Is this mean an accurate measure of the “average” earnings for these celebrities? (continued on next slide)

The Mean: Solution (a): Summing the salaries and dividing by 10 gives us 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.

What is the Median? Arrange the numbers in ascending order (smallest to greatest). 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? 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 = 73 divide by 2 = 36.5 million

The Mode:

The Mode: Example: Find the mode for each data set. a) The mode is 5. b) 3, 3, 3, 2, 1, 4, 4, 9, 9, 9 b) 2 modes: 3 and 9. c) 98, 99, 100, 101, 102 c) no mode. d) 2, 3, 4, 2, 3, 4, 5 d) no mode.

The Five Number Summary

The Five Number Summary 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. Find the following for this data set: a) the lower and upper halves b) the first and third quartiles c) the five-number summary (continued on next slide)

The Five Number Summary Solution: (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 (continued on next slide)

The Five Number Summary (c): The five number summary is We represent the five-number summary by a graph called a box-and-whisker plot. (continued on next slide)

Box & Whisker on TI-84 https://www.youtube.com/watch?v=iobQ-ge26kU Section 15.2, Slide 13

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