3-2 Measures of Variation 115 Heights: CV = s x # 100% = 7.10 cm 174.12 cm # 100% = 4.1% Weights: CV = s x # 100% = 17.65 kg 85.54 kg # 100% = 20.6% We can now see that the weights of adult males (with CV = 20.6%) vary considerably more than heights of adult males (with CV = 4.1%). YOUR TURN. Do Exercise 21 “Blood Pressure.” Measures of Variation Access tech supplements, videos, and data sets at www.TriolaStats.com Statdisk, Minitab, StatCrunch, Excel, R, and the TI-83>84 Plus Calculator can be used for the important calculations of this section. Use the same Descriptive Statistics procedures given at the end of Section 3-1 on page 98. TECH CENTER Statistical Literacy and Critical Thinking 1.Range Rule of Thumb for Estimating s The 153 heights of males from Data Set 1 “Body Data” in Appendix B vary from a low of 155.0 cm to a high of 193.3 cm. Use the range rule of thumb to estimate the standard deviation s and compare the result to the standard deviation of 7.10 cm calculated using the 153 heights. What does the result suggest about the accuracy of estimates of s found using the range rule of thumb? 3-2 Basic Skills and Concepts Biased and Unbiased Estimators The sample standard deviation s is a biased estimator of the population standard deviation s, which means that values of the sample standard deviation s do not tend to center around the value of the population standard deviation s. While individual values of s could equal or exceed s, values of s generally tend to underestimate the value of s. For example, consider an IQ test designed so that the population standard deviation is 15. If you repeat the process of randomly selecting 100 subjects, giving them IQ tests, and calculating the sample standard deviation s in each case, the sample standard deviations that you get will tend to be less than 15, which is the population standard deviation. There is no correction that allows us to fix the bias for all distributions of data. There is a correction that allows us to fix the bias for normally distributed populations, but it is rarely used because it is too complex and makes relatively minor corrections. The sample variance s2 is an unbiased estimator of the population variance s 2, which means that values of s2 tend to center around the value of s 2 instead of systematically tending to overestimate or underestimate s 2. Consider an IQ test designed so that the population variance is 225. If you repeat the process of randomly selecting 100 subjects, giving them IQ tests, and calculating the sample variance s2 in each case, the sample variances that you obtain will tend to center around 225, which is the population variance. Biased estimators and unbiased estimators will be discussed more in Section 6-3.
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