106 CHAPTER 3 Describing, Exploring, and Comparing Data Range Let’s begin with the range because it is quick and easy to compute, but it is not as important as other measures of variation. DEFINITION 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) Important Property of the Range ■ The range uses only the maximum and the minimum data values, so it is very sensitive to extreme values. The range is not resistant. ■ Because the range uses only the maximum and minimum values, it does not take every value into account and therefore does not truly reflect the variation among all of the data values. DEFINITION The standard deviation of a set of sample values, denoted by s, is a measure of how much data values deviate away from the mean. It is calculated by using Formula 3-4 or 3-5. Formula 3-5 is just a different version of Formula 3-4; both formulas are algebraically the same. The standard deviation found from sample data is a statistic denoted by s, but the standard deviation found from population data is a parameter denoted by s. The formula for s is slightly different with division by the population size N used instead of division by n - 1. The population standard deviation s will be discussed later. Notation s = sample standard deviation s = population standard deviation R L p Got a Second? The time unit of 1 second is defined to be “the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.” That definition redefines time to be based on the behavior of atoms instead of the earth’s motion. It results in accuracy of {1 second in 10,000,000 years, which is the most accurate measurement we use. Because it is so accurate, the second is being used to define other quantities, such as the meter. The meter was once defined as 1>10,000,000 of the distance along the surface of the earth between the North Pole and the equator (passing through Paris). The meter is now defined as the length of the distance traveled by light in a vacuum during a time interval of 1>299,792,458 sec. When dealing with time measurement devices, the traditional standard deviation has been found to be poor because of a trend in which the mean changes over time. Instead, other special measures of variation are used, such as Allan variance, total variance, and TheoH. Unrelated to statistics but nonetheless interesting is the fact that ads for watches usually show a watch with a time close to 10:10. That time allows the brand name to be visible, and it creates a subliminal image of a happy face. The time of 10:10 has been the industry standard since the 1940s. T o d “ 9 p r r t CP EXAMPLE 1 Range Find the range of these wait times (minutes) for Space Mountain. (These are the first eleven “Space Mountain” 10 AM wait times from Data Set 33 “Disney World Wait Times” in Appendix B.) 50 25 75 35 50 25 30 50 45 25 20 SOLUTION The range is found by subtracting the lowest value from the largest value, so we get Range = 1maximum value2 - 1minimum value2 = 75 - 20 = 55.0 minutes The range of 55.0 minutes is shown with one more decimal place than is present in the original data values. YOUR TURN. Find the range in Exercise 7 “Celebrity Net Worth.” Standard Deviation of a Sample The standard deviation is the measure of variation most commonly used in statistics.
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