110 CHAPTER 3 Describing, Exploring, and Comparing Data Variation in Faces Researchers commented that “if everyone looked more or less the same, there would be total chaos.” They studied human body measurements and found that facial traits varied more than other body traits, and the greatest variation occurred within the triangle formed by the eyes and mouth. They learned that facial traits vary independently of each other. For example, there is no relationship between the distance between your eyes and how big your mouth is. The researchers stated that our facial variation played an important role in human evolution. (See “Morphological and Population Genomic Evidence That Human Faces Have Evolved to Signal Individual Identity,” by Sheehan and Nachman, Nature Communications, Vol. 5, No. 4800.) CAUTION When using a calculator to find standard deviation, identify the notation used by your particular calculator so that you get the sample standard deviation, not the population standard deviation. Variance of a Sample and a Population So far, we have used the term variation as a general description of the amount that values vary among themselves. (The terms dispersion and spread are sometimes used instead of variation.) The term variance has a specific meaning. DEFINITION The variance of a set of values is a measure of variation equal to the square of the standard deviation. • Sample variance: s2 = square of the standard deviation s. • Population variance: s2 = square of the population standard deviation s. Standard Deviation of a Population The definition of standard deviation and Formulas 3-4 and 3-5 apply to the standard deviation of sample data. A slightly different formula is used to calculate the standard deviation s (lowercase sigma) of a population: Instead of dividing by n - 1, we divide by the population size N, as shown here: Population standard deviation s = CΣ1x - m2 2 N Because we generally deal with sample data, we will usually use Formula 3-4, in which we divide by n - 1. Many calculators give both the sample standard deviation and the population standard deviation, but they use a variety of different notations. EXAMPLE 5 Range Rule of Thumb for Estimating s Use the range rule of thumb to estimate the standard deviation of the sample of 147 pulse rates of females in Data Set 1 “Body Data” in Appendix B. Those 147 pulse rates have a minimum of 36 beats per minute and a maximum of 104 beats per minute. SOLUTION The range rule of thumb indicates that we can estimate the standard deviation by finding the range and dividing it by 4. With a minimum of 36 and a maximum of 104, the range rule of thumb can be used to estimate the standard deviation s, as follows: s ≈ range 4 = 104 - 36 4 = 17.0 beats per minute INTERPRETATION The actual value of the standard deviation is s = 12.5 beats per minute, so the estimate of 17.0 beats per minute is in the general neighborhood of the exact result. Because this estimate is based on only the minimum and maximum values, it is generally a rough estimate that could be off by a considerable amount. YOUR TURN. Do Exercise 29 “Estimating Standard Deviation.”
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