2. Excursions in Modern Mathematics, 7e: 14.3 - 2Copyright © 2010 Pearson Education, Inc. 14 Descriptive Statistics 14.1Graphical Descriptions of Data 14.2Variables 14.3 Numerical Summaries 14.4Measures of Spread

3. Excursions in Modern Mathematics, 7e: 14.3 - 3Copyright © 2010 Pearson Education, Inc. Measures of Location Measures of location such as the mean (or average), the median, and the quartiles, are numbers that provide information about the values of the data. Numerical Summaries of a Data Set

4. Excursions in Modern Mathematics, 7e: 14.3 - 4Copyright © 2010 Pearson Education, Inc. The best known of all numerical summaries of data is the average, also called the mean. There is no universal agreement as to which of these names is a better choice–in some settings mean is a better choice than average, in other settings it’s the other way around. In this chapter we will use whichever seems the better choice at the moment. Average or Mean

5. Excursions in Modern Mathematics, 7e: 14.3 - 5Copyright © 2010 Pearson Education, Inc. The average (or mean) of a set of N numbers is found by adding the numbers and dividing the total by N. In other words, the average of the numbers d 1, d 2, d 3,…, d N is A = (d 1 + d 2 + d 3 +…+ d N )/N. Average or Mean

6. Excursions in Modern Mathematics, 7e: 14.3 - 6Copyright © 2010 Pearson Education, Inc. In this example we will find the average test score in the Stat 101 exam first introduced in Example 14.1. To find this average we need to add all the test scores and divide by 75. The addition of the 75 test scores can be simplified considerably if we use a frequency table. Example 14.9Stat 101 Test Scores: Part 4

7. Excursions in Modern Mathematics, 7e: 14.3 - 7Copyright © 2010 Pearson Education, Inc. From the frequency table we can find the sum S of all the test scores as follows: Multiply each test score by its corresponding frequency and then add these products. Thus, the sum of all the test scores is S = (1  1) + (6  1) + (7  2) + (8  6) + …+ (16  1) + (24  1) = 814 If we divide this sum by N = 75, we get the average test score A = 814/75 ≈ 10.85 points. Example 14.9Stat 101 Test Scores: Part 4

8. Excursions in Modern Mathematics, 7e: 14.3 - 8Copyright © 2010 Pearson Education, Inc. In general, to find the average A of a data set given by a frequency table such as Table 14-8 we do the following: Step 1. S = d 1f 1 + d 2f 2 +… + d kf k To Find the Average Step 2. N = f 1 + f 2 +…+ f k Step 3. A = S/N

9. Excursions in Modern Mathematics, 7e: 14.3 - 9Copyright © 2010 Pearson Education, Inc. Imagine that you just read in the paper the following remarkable tidbit: The average starting salary of philosophy majors who recently graduated from Tasmania State University is $76,400 a year! This is quite an impressive number, but before we all rush out to change majors, let’s point out that one of the graduating philosophy majors happens to be basketball star “Hoops” Tallman, who is doing his thing in the NBA for a starting salary of $3.5 million a year. Example 14.10Starting Salaries of Philosophy Majors

10. Excursions in Modern Mathematics, 7e: 14.3 - 10Copyright © 2010 Pearson Education, Inc. If we were to take this one outlier out of the population of 75 philosophy majors, we would have a more realistic picture of what philosophy majors are making. Here is how we can do it. Example 14.10Starting Salaries of Philosophy Majors ■ The total of all 75 salaries is 75 times the average salary: 75  $76,400 = $5,730,000

11. Excursions in Modern Mathematics, 7e: 14.3 - 11Copyright © 2010 Pearson Education, Inc. ■ The total of the other 74 salaries (excluding Hoops’s cool 3.5 mill) is $5,730,000 – $3,500,000 = $2,230,000 ■ The average of the remaining 74 salaries is $2,230,000/74 ≈ $30,135 Example 14.10Starting Salaries of Philosophy Majors

12. Excursions in Modern Mathematics, 7e: 14.3 - 12Copyright © 2010 Pearson Education, Inc. Table 14-9 shows the monthly balance (monthly income minus monthly spending) in Billy’s budget over the past year. A negative amount indicates that Billy spent more than what he had coming in (adding to his credit card debt). Example 14.11Living Beyond Your Means

13. Excursions in Modern Mathematics, 7e: 14.3 - 13Copyright © 2010 Pearson Education, Inc. In spite of his consistent overspending, Billy’s average monthly balance for the year is $26 (check it out!). This average hides the true picture of what is going on. Billy is living well beyond his means but was bailed out by a lucky break and a generous mom. Example 14.11Living Beyond Your Means

14. Excursions in Modern Mathematics, 7e: 14.3 - 14Copyright © 2010 Pearson Education, Inc. While a single numerical summary–such as the average–can be useful, it is rarely sufficient to give a meaningful description of a data set. A better picture of the data set can be presented by using a well-organized cadre of numerical summaries. The most common way to do this is by means of percentiles. The pth percentile of a data set is a value such that p percent of the numbers fall at or below this value and the rest fall at or above it. It essentially splits a data set into two parts: the lower p% of the data values and the upper of the data values. Percentiles

15. Excursions in Modern Mathematics, 7e: 14.3 - 15Copyright © 2010 Pearson Education, Inc. There are several different ways to compute percentiles that will satisfy the definition,and different statistics books describe different methods. We will illustrate one such method. The first step is to sort the numbers from smallest to largest. Let’s denote the sorted data values by d 1, d 2, d 3, …, d N, where d 1 represents the smallest number in the data set, d 2 the second smallest number, and so on; d 3.5 represents the average of the data values d 3 and d 4, d 7.5 represents the average of the data values d 7 and d 8. Percentiles

16. Excursions in Modern Mathematics, 7e: 14.3 - 16Copyright © 2010 Pearson Education, Inc. The next, and most important, step is to identify which d represents the pth percentile of the data set. To do this, we compute the pth percent of N, which we will call the locator and denote by the letter L. [In other words, ] If L happens to be a whole number, then the pth percentile will be d L.5 (the average of d L and d L+1 ). If L is not a whole number, then the pth percentile will be d L+ where L + represents the value of L rounded up. Percentiles

17. Excursions in Modern Mathematics, 7e: 14.3 - 17Copyright © 2010 Pearson Education, Inc. ■ Step 0. Sort the data set from smallest to largest. Let d 1, d 2, d 3, …, d N represent the sorted data. ■ Step 1. Find the locator L = (p/100)N. FINDING THE pTH PERCENTILE OF A DATA SET

18. Excursions in Modern Mathematics, 7e: 14.3 - 18Copyright © 2010 Pearson Education, Inc. ■ Step 2. Depending on whether L is a whole number or not, the pth percentile is given by ■ d L.5 if L is a whole number. ■ d L+ if L is not a whole number (L + is L rounded up). FINDING THE pTH PERCENTILE OF A DATA SET

19. Excursions in Modern Mathematics, 7e: 14.3 - 19Copyright © 2010 Pearson Education, Inc. To reward good academic performance from its athletes, Tasmania State University has a program in which athletes with GPAs in the top 20th percentile of their team’s GPAs get a $5000 scholarship and athletes with GPAs in the top forty-fifth percentile of their team’s GPAs who did not get the $5000 scholarship get a $2000 scholarship. Example 14.12Scholarships by Percentiles

20. Excursions in Modern Mathematics, 7e: 14.3 - 20Copyright © 2010 Pearson Education, Inc. The women’s soccer team has N = 15 players. A list of their GPAs is as follows: 3.42, 3.91, 3.33, 3.65, 3.57, 3.45, 4.0, 3.71, 3.35, 3.82, 3.67, 3.88, 3.76, 3.41, 3.62 When we sort these GPAs we get the list 3.33, 3.35, 3.41, 3.42, 3.45, 3.57, 3.62, 3.65, 3.67, 3.71, 3.76, 3.82, 3.88, 3.91, 4.0 Example 14.12Scholarships by Percentiles

21. Excursions in Modern Mathematics, 7e: 14.3 - 21Copyright © 2010 Pearson Education, Inc. Since this list goes from lowest to highest GPA, we are looking for the 80th percentile and above (top 20th percentile) for the $5000 scholarships and the 55th percentile and above (top 45th percentile) for the $2000 scholarships. Example 14.12Scholarships by Percentiles

22. Excursions in Modern Mathematics, 7e: 14.3 - 22Copyright © 2010 Pearson Education, Inc. $5000 scholarships: The locator for the 80th percentile is (0.8)  15 = 12. Here the locator is a whole number, so the 80th percentile is given by d 12.5 = 3.85 (the average between d 12 = 3.82 and d 13 = 3.88). Thus, three students (the ones with GPAs of 3.88, 3.91 and 4.0) get $5000 scholarships. Example 14.12Scholarships by Percentiles

23. Excursions in Modern Mathematics, 7e: 14.3 - 23Copyright © 2010 Pearson Education, Inc. $2000 scholarships: The locator for the 55th percentile is (0.55)  15 = 8.25. Here the locator is not a whole number, so we round it up to 9, and the 55th percentile is given by d 9 = 3.67. Thus, the students with GPAs of 3.67, 3.71, 3.76 and 3.82 (all students with GPAs of 3.67 or higher except the ones that already received $5000 scholarships) get $2000 scholarships. Example 14.12Scholarships by Percentiles

24. Excursions in Modern Mathematics, 7e: 14.3 - 24Copyright © 2010 Pearson Education, Inc. The 50th percentile of a data set is known as the median and denoted by M. The median splits a data set into two halves–half of the data is at or below the median and half of the data is at or above the median. Median

25. Excursions in Modern Mathematics, 7e: 14.3 - 25Copyright © 2010 Pearson Education, Inc. ■ Sort the data set from smallest to largest. Let d 1, d 2, d 3, …, d N represent the sorted data. ■ If N is odd, the median is ■ If N is even, the median is the average of FINDING THE MEDIAN OF A DATA SET

26. Excursions in Modern Mathematics, 7e: 14.3 - 26Copyright © 2010 Pearson Education, Inc. After the median, the next most commonly used set of percentiles are the first and third quartiles. The first quartile (denoted by Q 1 ) is the 25th percentile, and the third quartile (denoted by Q 3 ) is the 75th percentile. Quartiles

27. Excursions in Modern Mathematics, 7e: 14.3 - 27Copyright © 2010 Pearson Education, Inc. During the last year, 11 homes sold in the Green Hills subdivision. The selling prices, in chronological order, were $267,000, $252,000, $228,000, $234,000, $292,000, $263,000, $221,000, $245,000, $270,000, $238,000, and $255,000. We are going to find the median and the quartiles of the N = 11 home prices. Example 14.13Home Prices in Green Hills

28. Excursions in Modern Mathematics, 7e: 14.3 - 28Copyright © 2010 Pearson Education, Inc. Sorting the home prices from smallest to largest (and dropping the 000’s) gives the sorted list 221, 228, 234, 238, 245, 252, 255, 263, 267, 270, 292 Example 14.13Home Prices in Green Hills

29. Excursions in Modern Mathematics, 7e: 14.3 - 29Copyright © 2010 Pearson Education, Inc. The locator for the median is (0.5)  11 = 5.5, the locator for the first quartile is (0.25)  11 = 2.75, and the locator for the third quartile is (0.75)  11 = 8.25. Since these locators are not whole numbers, they must be rounded up: 5.5 to 6, 2.75 to 3, and 8.25 to 9. Example 14.13Home Prices in Green Hills

30. Excursions in Modern Mathematics, 7e: 14.3 - 30Copyright © 2010 Pearson Education, Inc. Thus, the median home price is given by d 6 = 252 (i.e., M = $252,000), the first quartile is given by d 3 = 234 (i.e., M = $234,000), and the third quartile is given by d 9 = 267 (i.e., M = $267,000). Example 14.13Home Prices in Green Hills

31. Excursions in Modern Mathematics, 7e: 14.3 - 31Copyright © 2010 Pearson Education, Inc. Oops! Just this morning a home sold in Green Hills for $264,000. We need to recalculate the median and quartiles for what are now N = 12 home prices. We can use the sorted data set that we already had–all we have to do is insert the new home price (264) in the right spot (remember, we drop the 000’s!). This gives 221, 228, 234, 238, 245, 252, 255, 263, 264, 267, 270, 292 Example 14.13Another Home Sells in Green Hills

32. Excursions in Modern Mathematics, 7e: 14.3 - 32Copyright © 2010 Pearson Education, Inc. Now N = 12 and in this case the median is the average of d 6 = 252 and d 7 = 255. It follows that the median home price is M = $253,500. The locator for the first quartile is (0.25)  12 = 3, since the locator is a whole number, the first quartile is the average of d 3 = 234 and d 4 = 238 (i.e., Q 1 = $236,000). Similarly, the third quartile is Q 3 = $265,500 (the average of d 9 = 264 and d 10 = 267). Example 14.13Another Home Sells in Green Hills

33. Excursions in Modern Mathematics, 7e: 14.3 - 33Copyright © 2010 Pearson Education, Inc. We will now find the median and quartile scores for the Stat 101 data set (shown again in Table 14-10). Having the frequency table available eliminates the need for sorting the scores–the frequency table has, in fact, done this for us. Example 14.14Stat 101 Test Scores: Part 5

34. Excursions in Modern Mathematics, 7e: 14.3 - 34Copyright © 2010 Pearson Education, Inc. Here N = 75 (odd), so the median is the thirty- eighth score (counting from the left) in the frequency table. To find the thirty-eighth number in Table 14-10, we tally frequencies as we move from left to right: 1 + 1= 2; 1 + 1 + 2 = 4; 1 + 1 + 2 + 6 = 10; 1 + 1 + 2 + 6 + 10 = 20; 1 + 1 + 2 + 6 + 10 + 16 = 36. At this point, we know that the 36th test score on the list is a 10 (the last of the 10’s) and the next 13 scores are all 11’s. We can conclude that the 38th test score is 11. Thus, M = 11. Example 14.14Stat 101 Test Scores: Part 5

35. Excursions in Modern Mathematics, 7e: 14.3 - 35Copyright © 2010 Pearson Education, Inc. The locator for the first quartile is L = (0.25)  75 = 18.75. Thus, Q 1 = d 19. To find the nineteenth score in the frequency table, we tally frequencies from left to right: 1 + 1 = 2; 1 + 1 + 2 = 4; 1 + 1 + 2 + 6 = 10; 1 + 1 + 2 + 6 + 10 = 20. At this point we realize that d 10 = 8 (the last of the 8’s) and that d 11 through d 10 all equal 9. Hence, the first quartile of the Stat 101 midterm scores is Q 1 = d 19 = 9. Example 14.14Stat 101 Test Scores: Part 5

36. Excursions in Modern Mathematics, 7e: 14.3 - 36Copyright © 2010 Pearson Education, Inc. Since the first and third quartiles are at an equal “distance” from the two ends of the sorted data set, a quick way to locate the third quartile now is to look for the nineteenth score in the frequency table when we count frequencies from right to left. We find the third quartile of the Stat 101 data set is Q 3 = 12. Example 14.14Stat 101 Test Scores: Part 5

37. Excursions in Modern Mathematics, 7e: 14.3 - 37Copyright © 2010 Pearson Education, Inc. In this example we continue the discussion of the 2007 SAT math scores introduced in Example 14.6. Recall that the number of college-bound high school seniors taking the test was N = 1,494,531. As reported by the College Board, the median score in the test was M = 510, the first quartile score was Q 1 =430, and the third quartile was Q 3 = 590. What can we make of this information? Example 14.152007 SAT Math Scores: Part 2

38. Excursions in Modern Mathematics, 7e: 14.3 - 38Copyright © 2010 Pearson Education, Inc. Let’s start with the median. From N = 1,494,531 (an odd number), we can conclude that the median (510 points) is the 747,266th score in the sorted list of test scores. This means that there were at least 747,266 students who scored 510 or less in the math section of the 2007 SAT. Why did we use “at least” in the preceding sentence? Example 14.152007 SAT Math Scores: Part 2

39. Excursions in Modern Mathematics, 7e: 14.3 - 39Copyright © 2010 Pearson Education, Inc. Could there have been more than that number who scored 510 or less? Yes, almost surely. Since the number of students who scored 510 is in the thousands, it is very unlikely that the 747,266th score is the last of the 510s. In a similar vein, we can conclude that there were at least 373,633 scores of Q 1 = 430 or less [the locator for the first quartile is (0.25)  1,494,531 = 373,632.75] and at least 1,120,899 scores of Q 3 = 590 or less. Example 14.152007 SAT Math Scores: Part 2

40. Excursions in Modern Mathematics, 7e: 14.3 - 40Copyright © 2010 Pearson Education, Inc. Medians, quartiles, and general percentiles are often computed using statistical calculators or statistical software packages, which is all well and fine since the whole process can be a bit tedious. The problem is that there is no universally agreed upon procedure for computing percentiles, so different types of calculators and different statistical packages may give different answers from each other and from those given in this book for quartiles and other percentiles (everyone agrees on the median). A Note of Warning

41. Excursions in Modern Mathematics, 7e: 14.3 - 41Copyright © 2010 Pearson Education, Inc. A common way to summarize a large data set is by means of its five-number summary. The five-number summary is given by (1) the smallest value in the data set (called the Min), (2) the first quartile Q 1, (3) the median M, (4) the third quartile Q 3, and (5) the largest value in the data set (called the Max). These five numbers together often tell us a great deal about the data. The Five-Number Summary

42. Excursions in Modern Mathematics, 7e: 14.3 - 42Copyright © 2010 Pearson Education, Inc. For the Stat 101 data set, the five-number summary is Min = 1, Q 1 = 9, M = 11, Q 3 = 12, Max = 24. What useful information can we get out of this? Right away we can see that the N = 75 test scores were not evenly spread out over the range of possible scores. For example, from M = 11 and Q 3 = 12 we can conclude that at least 25% of the class (that means at least 19 students) scored either 11 or 12 on the test. Example 14.16Stat 101 Test Scores: Part 6

43. Excursions in Modern Mathematics, 7e: 14.3 - 43Copyright © 2010 Pearson Education, Inc. At the same time, from Q 3 = 12 and Max = 24 we can conclude that less than one-fourth of the class (i.e., at most 18 students) had scores in the 13–24 point range. Using similar arguments, we can conclude that at least 19 students had scores between Q 1 = 9 and M = 11 points and no more than 18 students scored in the 1–8 point range. Example 14.16Stat 101 Test Scores: Part 6

44. Excursions in Modern Mathematics, 7e: 14.3 - 44Copyright © 2010 Pearson Education, Inc. The “big picture” we get from the five-number summary of the Stat 101 test scores is that there was a lot of bunching up in a narrow band of scores (at least half of the students in the class scored in the range 9–12 points), and the rest of the class was all over the place. In general, this type of “bumpy” distribution of test scores is indicative of a test with an uneven level of difficulty–a bunch of easy questions and a bunch of really hard questions with little in between. Example 14.16Stat 101 Test Scores: Part 6

45. Excursions in Modern Mathematics, 7e: 14.3 - 45Copyright © 2010 Pearson Education, Inc. Invented in 1977 by statistician John Tukey, a box plot (also known as a box-and- whisker plot) is a picture of the five-number summary of a data set. The box plot consists of a rectangular box that sits above a scale and extends from the first quartile Q 1 to the third quartile Q 3 on that scale. A vertical line crosses the box, indicating the position of the median M. On both sides of the box are “whiskers” extending to the smallest value, Min, and largest value, Max, of the data. Box Plots

46. Excursions in Modern Mathematics, 7e: 14.3 - 46Copyright © 2010 Pearson Education, Inc. This figure shows a generic box plot for a data set. Box Plots

47. Excursions in Modern Mathematics, 7e: 14.3 - 47Copyright © 2010 Pearson Education, Inc. This figure shows a box plot for the Stat 101 data set. The long whiskers in this box plot are largely due to the outliers 1 and 24. Box Plots

48. Excursions in Modern Mathematics, 7e: 14.3 - 48Copyright © 2010 Pearson Education, Inc. This figure shows a variation of the same box plot, but with the two outliers, marked with two crosses, segregated from the rest of the data. Box Plots

49. Excursions in Modern Mathematics, 7e: 14.3 - 49Copyright © 2010 Pearson Education, Inc. This figure shows box plots for the starting salaries of two different populations: first-year agriculture and engineering graduates of Tasmania State University. Example 14.17Comparing Agriculture and Engineering Salaries

50. Excursions in Modern Mathematics, 7e: 14.3 - 50Copyright © 2010 Pearson Education, Inc. Superimposing the two box plots on the same scale allows us to make some useful comparisons. It is clear, for instance, that engineering graduates are doing better overall than agriculture graduates, even though at the very top levels agriculture graduates are better paid. Example 14.17Comparing Agriculture and Engineering Salaries

51. Excursions in Modern Mathematics, 7e: 14.3 - 51Copyright © 2010 Pearson Education, Inc. Another interesting point is that the median salary of agriculture graduates ($43,000) is less than the first quartile of the salaries of engineering graduates ($45,000). Example 14.17Comparing Agriculture and Engineering Salaries

52. Excursions in Modern Mathematics, 7e: 14.3 - 52Copyright © 2010 Pearson Education, Inc. The very short whisker on the left side of the agriculture box plot tells us that the bottom 25% of agriculture salaries are concentrated in a very narrow salary range ($32,500– $35,000). Example 14.17Comparing Agriculture and Engineering Salaries

53. Excursions in Modern Mathematics, 7e: 14.3 - 53Copyright © 2010 Pearson Education, Inc. We can also see that agriculture salaries are much more spread out than engineering salaries,even though most of the spread occurs at the higher end of the salary scale. Example 14.17Comparing Agriculture and Engineering Salaries