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Chapter 3 Data Summary Using Descriptive Measures

KVANLI PAVUR KEELING. Chapter 3 Data Summary Using Descriptive Measures. Chapter Objectives. At the completion of this chapter, you should be able to define and use the following measures: ∙ Measures of Central Tendency: Mean, Median, Mode and Midrange

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Chapter 3 Data Summary Using Descriptive Measures

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  1. KVANLI PAVUR KEELING Chapter 3Data Summary Using Descriptive Measures

  2. Chapter Objectives • At the completion of this chapter, you should be able to define and use the following measures: ∙ Measures of Central Tendency: Mean, Median, Mode and Midrange ∙ Measures of Variation: Range, Standard Deviation, Variance, and Coefficient of Variation

  3. Chapter Objectives - Continued • At the completion of this chapter, you should be able to define and use the following measures: ∙ Measures of Position: Percentiles, Quartiles, and z-scores ∙ Measures of Shape: Skewness and Kurtosis

  4. Summarizing a Sample • Chapter 2 described a sample using a graph or chart • This chapter summarizes a sample by crunching a number or two, such as an average • We refer to these number as descriptivemeasures • There are four different types of descriptive measures

  5. Descriptive Measures • There are measures of: • central tendency • variation • position • shape • Consider a sample consisting of the number of purchased textbooks this semester for 5 randomly selected students • The sample values are {6, 9, 7, 23, 5} Here, the sample size is n = 5

  6. Measures of Central Tendency • These are: • mean • median • midrange • mode These determine where the “middle” of the sample is; that is, a “typical” value The mode is that value that occurs the most often

  7. Example • Consider a sample consisting of the number of purchased textbooks this semester for 5 randomly selected students • The sample values are {6, 9, 7, 23, 5} Here, the sample size is n = 5

  8. The Sample Mean • The sample mean is the sample average • Our sample: {6, 9, 7, 23, 5} • The sample mean is books • The symbol for the sample mean is • So, = 10 Read as “x bar”

  9. The Sample Median • Median is the center of the ordered array • Two steps are involved here: • Step 1: • Order the data from smallest to largest • For our sample, this would be {5, 6, 7, 9, 23} • Step 2: • For an odd number of data values in the distribution, Median = Middle data value of the ordered data Here, Md = 7 • In general for n odd, Md is the value Here, this would be the 3rd value

  10. The Sample Median – n is Even • For an even number of data values in the distribution, Median = Sum of the middle two values 2 • Consider this sample: {2, 4, 8, 12, 16, 18} (n = 6) • Here, Md = books • In general for n even, Md is the average of the value and the next one

  11. The Sample Midrange • The midrange is the average of lowest (L) and highest (H) sample values • The symbol for the midrange is Mr • Mr= • The textbook sample is {6, 9, 7, 23, 5} • Here, Mr = This is H This is L

  12. The Sample Mode • The mode (Mo) is that value that occurs the most often in the sample • For the textbook example, there is no mode since there are no repeat values • If there is a 2-way tie, you state that the modes are ____ and ____ • For continuous data, don’t bother looking for a mode

  13. More on the Sample Mode • If your company manufactures clothing, the sample mode is more likely to be of interest rather than the other three measures of central tendency • Example: You company manufactures hats • The statistic of interest in a sample of head sizes would be the most popular head size since we should manufacture more hats of that size • The mean (say, 6.82) would be of little interest • Ditto for the median and midrange

  14. Outliers • In statistics, Outliers in a data set are data values that are very different from other measurements in the data set. • Consider the sample {5, 6, 7, 9, 54} • 54 is called an outliersince it is unusually large and doesn’t fit with the other four values • When trying to determine the middle (a “typical value”), which of these three measures of central tendency ( , Md & Mr)were most affected by this outlier?

  15. The Effect of an Outlier • This outlier had the biggest impact on the midrange • This outlier also had a big impact on the mean • But the outlier had NO effect on the sample median

  16. Outliers and the Median • To illustrate this, suppose the sample values are {5, 6, 7, 9, 54} • The midrange and mean are considerably larger than before (Mr = 24.5, = 16.2 ) • But the sample median is still 7 • It didn’t even change!

  17. Moral to the Story • If you expect (or know) your sample contains outliers, use the median. Otherwise, use the mean. • Examples Incomes usually contain a few very large values. Use the median. House prices in a particular neighborhood typically contain a few very large values. Use the median.

  18. Calculators • Most any calculator will work in this course. • If you prefer to use the TI-83 or TI-86, there are links on the DSCI 2710 website that show you how to crunch numbers on these two calculators. • If you’re going to purchase a calculator, I’d recommend the TI BA II Plus. It works very well in this course and is easy to use.

  19. Measures of Variation • Variation are measures to determine how much the sample values jump around the mean. • These are: • range (R) • variance (s2) • standard deviation (s) • coefficient of variation (CV) The most popular

  20. The Sample Range (R) • The range is the difference of the highest and lowest sample values of a data distribution • R = H – L • Textbook sample: {5, 6, 7, 9, 23} • R = 23 – 5 = 18 • The sample range is a good measure of variation (and easy to compute) for small samples ( n ≤ 10)

  21. The Sample Variance x 5 5 – 10 = -5 25 6 6 – 10 = -4 16 7 7 – 10 = -3 9 9 9 – 10 = -1 1 23 23 – 10 = 13169 0 220 Always is

  22. The Sample Variance • The sum of the squared deviations (220) is then divided by n – 1 (not the sample size (n) as you might expect) • This is the sample variance (s2) • s2 = • In general, s2 =

  23. The Sample Standard Deviation • The sample standard deviation (s) is the square root of the variance • s = • Here, s = • The units on the standard deviation are the same as the units on the sample data • For this example, s = 7.416 books

  24. Using a Calculator • When deriving the standard deviation, we first found the variance and then found the square root of this value • When using a calculator, you reverse this sequence by ∙ entering the sample values ∙ hit the standard deviation key ∙ if you want a variance, hit the x2 key This is the way to do it!

  25. The Sample Coefficient of Variation • The coefficient of variation (CV) is useful for comparing variation in two or more samples • Consider the following two samples: Sample #1 {5, 6, 7, 9, 23} the textbook sample Sample #2 {500, 600, 700, 900, 2300} • Question: Which sample has more variation?

  26. The Sample Coefficient of Variation • For sample #1: x = 10 and s = 7.416 • For sample #2, it turns out that the previous mean and standard deviation are simply multiplied by 100; that is, x = 1000 and s = 741.6 • If you compared the two standard deviations, you might conclude that sample #2 has more variation • But, these sample values are simply larger

  27. The Sample Coefficient of Variation • In fact, relative to the mean, the variation in these two samples is the same • To effectively compare the variation in these two samples, you should compute the coefficient of variation (CV) for each sample, where CV = · 100

  28. The Sample Coefficient of Variation • For sample #1, CV = · 100 = 74.16 • For sample #2, CV = · 100 = 74.16 • For both samples, the standard deviation is 74.16% of the mean • MORAL: Don’t compare sample standard deviations unless the sample means are about the same

  29. Describing a Population • Populations also have a mean, variance, and standard deviation Population (size is N) The population mean is μ (mu, pronounced “myoo”) and the population standard deviation is σ (sigma) The sample mean is and the sample standard deviation is s Sample (size is n)

  30. The Dreaded Formulas Sample Population You divide by N (not N-1) σ2 is the population variance

  31. Question Belleview College must make a report to the budget committee about the average credit hour load a full-time student carries. ( A 12-credit-hour load is the minimum requirement for full-time status. For the same tuition, students may take up to 20 credit hours.) A random sample of 20 students yielded the following information (in credit hours): 15 12 15 16 12 18 20 19 12 15 18 14 16 17 15 19 12 13 12 15 • Find the median credit hour load. • Find the modal credit hour load.

  32. Question • Cabela’s in Sidney, Nebraska, is a very large outfitter that carries a broad selection of fishing tackle. It markets its products nationwide through a catalog service. A random sample of 10 spinners from Cabela’s extensive spring catalog gave the following prices (in dollars): 1.69 1.49 3.09 1.79 1.39 2.89 1.49 1.39 1.49 1.99 Compute the CV for the spinner prices at Cabela’s.

  33. Question Big Blossom Greenhouse was commissioned to develop an extra large rose for the Rose Bowl Parade. A random sample of blossoms from Hybrid A bushes yielded the following diameters (in inches) for mature peak blooms. 2 3 3 8 10 10 Find the sample variance and standard deviation.

  34. Question Certain kinds of tumors tend to recur. The following data represent the lengths of time, in months, for a tumor to recur after chemotherapy. Using five classes, construct a frequency distribution table (class number, class, frequency and relative frequency) for the following data set. 19 18 17 1 21 22 54 46 25 49 50 1 59 39 43 39 5 9 38 18 14 45 54 46 50 29 12 19 36 38 40 43 41 10 50 41 25 19 39 27 20 59

  35. Measures of Position • Measures of position include • Percentiles ∙ Special percentiles • z-scores These are called quartiles

  36. Percentiles (the most common measure of position) • Consider the 50 aptitude test scores introduced in Chapter 2 These values must be sorted Table 3.2

  37. Percentiles • There are two rules to apply here • What is the 35th percentile? This uses Rule #1 • To find: 50 · .35 = 17.5 • Rule #1: Round this product up (always up) • So, the 35th percentile is the 18th value in the ordered array Rule #1 applies when this is not a counting number

  38. The 35th Percentile

  39. Percentiles • What is the 60th percentile? This uses Rule #2 • To find: 50 · .60 = 30 • Rule #2: The percentile is the average of this value (the 30th value) and the next one (the 31st value) Rule #2 applies when this is a counting number

  40. The 60th Percentile

  41. Quartiles • These are special percentiles • There are three quartiles (Q1, Q2, and Q3) • Q1 is the 25th percentile • Q2 is the 50th percentile • Q3 is the 75th percentile

  42. Quartiles • Using the 50 aptitude scores, determine Q1 • This is the 25th percentile: 50 · .25 = 12.5 • So, Q1 is the 13th value in the ordered array • This value is 46 Not a counting number. So, use Rule #1

  43. Quartiles • Q2 is the 50th percentile: 50 · .50 = 25 • Q2 is the average of the 25th and the 26th values in the ordered array • Q2 = (61 + 63)/2 = 62 • This is also the sample median • These two rules guarantee that Q2 is always equal to the sample median (and the 50th percentile) This is a counting number. So, use Rule #2

  44. Quartiles • Q3 is the 75th percentile: 50 · .75 = 37.5 • So, Q3 is the 38th value in the ordered array • This value is 75 Not a counting number. So, use Rule #1 It was just a fluke that the 75th percentile was equal to 75

  45. Another Measure of Position • A z-score is another measure of position • Every value in your sample has a corresponding z-score • How to find: z-score = where x is the sample value, x is the sample mean, and s is the sample standard deviation • The value of a z-score is how many standard deviations that sample value is to the left or right of the sample mean

  46. Finding a z-score =60.36 s = 18.61

  47. Finding a z-score • The corresponding z-score is • 90 is 1.59 standard deviations to the right of the mean • A z-score is positive if the sample value lies to the right of the mean and is negative if the sample value lies to the left of the mean • Typically, about half the z-scores will be positive and about half will be negative

  48. Question Let us consider these sets of data from a ransom sample of blossoms from Hybrid A bushes: • 3 3 8 10 10 Find the corresponding z-scores for sample values 3 and 8.

  49. Interpreting a z-score • These results are usually true Your z-score is 2.3 Approx. 68% z-score -3 -2 -1 0 1 2 3 Approx. 95% Nearly all

  50. Interpreting a z-score - Assumptions • The previous slide is called the Empirical Rule • This rule assumes that the population from which you got the sample is bell-shaped • This means that if you were able to get the entire population and make a histogram of it, it would resemble the histogram on the next slide. • This is generally (approximately) true – but not always

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