MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3 1 Review and Preview 3 2 Measures of Center 3 3 Measures of Variation 3 4 Measures of Relative Standing and Boxplots Key Concept Discuss characteristics of variation, in particular, measures of variation, such as standard deviation, for analyzing data. Make understanding and interpreting the standard deviation a priority. The range of a set of data values is the difference between the maximum data value and the minimum data value. Range = ( maximum data value) ( minimum data value) It is very sensitive to extreme values; therefore not as useful as other measures of variation. Round Off Rule for Measures of Variation When rounding the value of a measure of variation, carry one more decimal place than is present in the original set of data. Round only the final answer, not values in the middle of a calculation. The standard deviation of a set of sample values, denoted by s, is a measure of variation of values about the mean. It is a type of average deviation of values from the mean that is calculated by using Formula 3 4 or 3 5. Formula 3 5 is just a different version of Formula 3 4; it is algebraically the same. Formula 3 4: Sample Standard Deviation Formula 3 5: Shortcut formula for Sample Standard Deviation 1
Standard Deviation Important Properties The standard deviation is a measure of variation of all values from the mean. The value of the standard deviation s is usually positive. The value of the standard deviation s can increase dramatically with the inclusion of one or more outliers (data values far away from all others). The units of the standard deviation s are the same as the units of the original data values. Range Rule of Thumb The standard deviation, s, is approximately equal to the range divided by 4. Comparing Variation in Different Samples It s a good practice to compare two sample standard deviations only when the sample means are approximately the same. When comparing variation in samples with very different means, it is better to use the coefficient of variation, which is defined later in this section. Population Standard Deviation Rationale for using n 1 versus n There are only n 1 independent values. With a given mean, only n 1 values can be freely assigned any number before the last value is determined. Dividing by n 1 yields better results than dividing by n. It causes s 2 to target σ 2 whereas division by n causes s 2 to underestimate σ 2. This formula is similar to the previous formula, but instead, the population mean and population size are used. 2
Empirical rule. This rule states that for data sets having a distribution that is approximately bell shaped, the following properties apply. (See Figure 3 3.) About 68% of all values fall within 1 standard deviation of the mean. About 95% of all values fall within 2 standard deviations of the mean. About 99.7% of all values fall within 3 standard deviations of the mean. Chebyshev s Theorem The proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at least 1 1/ K 2, where K is any positive number greater than 1. For K = 2 and K = 3, we get the following statements: At least 3/4 (or 75%) of all values lie within 2 standard deviations of the mean. At least 8/9 (or 89%) of all values lie within 3 standard deviations of the mean. The coefficient of variation (or CV) for a set of nonnegative sample or population data, expressed as a percent, describes the standard deviation relative to the mean, and is given by the following: Sample Population In Exercises 5 20, find the (a) range, (b) variance, and (c) standard deviation for the given sample data. Include appropriate units (such as minutes ) in your results. (The same data were used in Section 3 2 where we found measures of center. Here we find measures of variation.) Then answer the given questions. 116/6. Tests of Child Booster Seats The National Highway Traffic Safety Administration conducted crash tests of child booster seats for cars. Listed below are results from those tests, with the measurements given in hic ( standard head injury condition units). According to the safety requirement, the hic measurement should be less than 1000. Do the different child booster seats have much variation among their crash test measurements? 774 649 1210 546 431 612 3
116/8. FICO Scores A simple random sample of FICO credit rating scores is listed below. As of this writing, the mean FICO score was reported to be 678. Based on these results, is a FICO score of 500 unusual? Why or why not? 714 751 664 789 818 779 698 836 753 834 693 802 117/17. Years to Earn Bachelor s Degree Listed below are the lengths of time ( in years) it took for a random sample of college students to earn bachelor s degrees ( based on data from the U. S. National Center for Education Statistics). Based on these results, is it unusual for someone to earn a bachelor s degree in 12 years? 4 4 4 4 4 4 4.5 4.5 4.5 4.5 4.5 4.5 6 6 8 9 9 13 13 15 117/19. Bankruptcies Listed below are the numbers of bankruptcy filings in Dutchess County, New York State. The numbers are listed in order for each month of a recent year ( based on data from the Poughkeepsie Journal ). Identify any of the values that are unusual. 59 85 98 106 120 117 97 95 143 371 14 15 118/22. In Exercises 21 24, find the coefficient of variation for each of the two sets of data, then compare the variation. (The same data were used in Section 3 2.) BMI for Miss America The trend of thinner Miss America winners has generated charges that the contest encourages unhealthy diet habits among young women. Listed below are body mass indexes ( BMI) for Miss America winners from two different time periods. BMI for 1920s & 1930s: 20.4 21.9 22.1 22.3 20.3 18.8 18.9 19.4 18.4 19.1 BMI for recent years: 19.5 20.3 19.6 20.2 17.8 17.9 19.1 18.8 17.6 16.8 4
118/24. In Exercises 21 24, find the coefficient of variation for each of the two sets of data, then compare the variation. (The same data were used in Section 3 2.) Customer Waiting Times Waiting times (in minutes) of customers at the Jefferson Valley Bank (where all customers enter a single waiting line) and the Bank of Providence ( where customers wait in individual lines at three different teller windows) are listed below. Jefferson Valley ( single line): 6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7 Providence ( individual lines): 4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0 119/29. Finding Standard Deviation from a Frequency Distribution. In Exercises 29 and 30, find the standard deviation of sample data summarized in a frequency distribution table by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 3 4 with the original list of data values: (Exercise 29) 3.2 mg; (Exercise 30) 12.5 beats per minute. 119/30. Finding Standard Deviation from a Frequency Distribution. In Exercises 29 and 30, find the standard deviation of sample data summarized in a frequency distribution table by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 3 4 with the original list of data values: (Exercise 29) 3.2 mg; (Exercise 30) 12.5 beats per minute. 5