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14 Descriptive Statistics What a Data Set Tells Us Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section Section14.2, 1.1, Slide Slide11 14.2 Measures of Central Tendency • Compute the mean, median, and mode of distributions. • Find the five-number summary of a distribution. • Apply measures of central tendency to compare data. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 2 The Mean and the Median 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 3 The Mean and the Median • 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 4 The Mean and the Median • 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? (continued on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 5 The Mean and the Median • Solution: A third column is added to the table that contains the products of the raw scores and their frequencies. The mean is Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 6 The Mean and the Median Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 7 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? (continued on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 8 The Mean and the Median • 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 9 The Mean and the Median Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 10 The Mean and 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 11 The Mean and the Median • 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? Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 12 The Mean and the Median • 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 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 13 The Five Number Summary Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 14 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 halves b) the first and third quartiles c) the five-number summary (continued on next slide) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 15 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) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 16 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) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 17 The Five Number Summary Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 18 The Five Number Summary • Example: Find the mode for each data set. a) 5, 5, 68, 69, 70 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 • Solution: a) The mode is 5. b) There are two modes: 3 and 9. In c) and d) there is no mode. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 19 Comparing Measures of Central Tendency • 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? Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 20 Comparing Measures of Central Tendency • 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. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 14.2, Slide 21