The Range Rule of Thumb is based on the insight that the range is often approximately 4 times as large as the standard deviation. This insight enables you to quickly estimate the standard deviation of a data set by finding the maximum and minimum data values, using them to compute the range, and then dividing this value by 4.
Each of the following 5 data sets has a mean close to 100, and a standard deviation in the 20s. Assume that each of these are sample data sets. For each of the 5 data sets given below, use the Excel function “=stdev.s” (or technology) to find the actual sample standard deviation (10 points), compute the estimate given by the Range Rule of Thumb (RROT) Estimate (10 points), and compute the percentage error of the estimate (10 points). The formula for the RROT estimate is RROT estimate = Range/4 (where the range = maximum value – minimum value), on the assumption that the Standard Deviation RROT estimate. The formula used to find the percentage error of the estimate is:
% Error = | RROT Estimate – Standard deviation | / Standard Deviation × 100 = ____ % .
Data Set 1: No outliers
68 68 68 68 68 133 133 133 133 133
Actual Standard deviation = RROT Estimate = % Error =
Data Set 2: 1 outlier
68 81 81 81 81 82 82 82 82 160
Actual Standard deviation = RROT Estimate = % Error =
Data Set 3: 2 outliers
38 86 88 89 90 90 91 92 104 152
Actual Standard deviation = RROT Estimate = % Error =
Data Set 4: Normal (Bell-Shaped) Distribution
51 67 74 81 87 93 99 106 115 128
Actual Standard deviation = RROT Estimate = % Error =
Data Set 5: Uniform Distribution (evenly-spaced values)
65 73 81 89 97 105 113 121 129 137
Actual Standard deviation = RROT Estimate = % Error =
Summarize what you learned:
a) Does the Range Rule of Thumb estimate more often underestimate or overestimate the actual standard deviation? (3 points)
b) What is the largest percentage error? (3 points)
c) Each of the data sets given above is artificially contrived. If you were to apply the RROT estimate to a typical real-world data set, about how accurate would you expect it to be (your answer here should be a numerical estimate of the percentage error, based on your previous results)? (4 points)
