Section 3-3 Measures of Variation Part B Created by Tom Wegleitner, Centreville, Virginia Slide 2 1 In a data list, every value falls within some number of standard deviations from the mean. Slide 3
Find the mean and standard deviation of the data for daily energy demand in a small town during August. Daily Energy Demand During August (MWh) Sun. Mon. Tues. Wed. Thur. Fri. Sat. 53 52 47 50 39 33 40 41 44 47 49 43 39 47 49 54 53 46 36 33 45 45 42 43 39 33 33 40 40 41 42 Slide 4 2 Estimation of Standard Deviation Range Rule of Thumb For estimating a value of the standard deviation s, Use Range s 4 Where range = (maximum value) (minimum value) Slide 5
Estimation of Standard Deviation Range Rule of Thumb For interpreting a known value of the standard deviation s, find rough estimates of the minimum and maximum usual sample values by using: Minimum usual value Maximum usual value = = (mean) 2 X (standard deviation) (mean) + 2 X (standard deviation) Slide 6 3 Example: Ages of Best Actresses Use the range rule of thumb to find a rough estimate of the standard deviation of the sample of 76 ages of actresses who won Oscars in the category of Best Actress. Correct value: s = 11.1 The estimate is a crude estimate that is off by a considerable amount. Slide 7
Example: Pulse Rates of Women Past results from the National Health Survey suggest that the pulse rates (beats per minute) have a mean of 76.0 and a standard deviation of 12.5. Use the range rule of thumb to find minimum and maximum usual pulse rates. (These results could be used by a physician who can identify unusual pulse rates that might be the result of some disorder.) Then determine whether a pulse rate of 110 would be considered unusual Minimum usual value = (mean) 2 x (standard deviation) = 76.0 2(12.5) = 51 beats per minute Maximum usual value = (mean) + 2 x (standard deviation) = 76.0 + 2(12.5) = 101 beats per minute Interpretation: Based on these results, we expect typical women have pulse rates between 51 and 101 beats per minute. Because 110 beats minute does not fall within those limits, it would be considered unusual With a pulse rate of 110, a physician might try to establish a reason for This unusual reading. 4 Slide 8 Definition Empirical (68-95-99.7) Rule For data sets having a distribution that is approximately bell shaped, the following properties apply: 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. Slide 9
The Empirical Rule Slide 10 5 The Empirical Rule Slide 11
The Empirical Rule Slide 12 6 Definition 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, at least 3/4 (or 75%) of all values lie within 2 standard deviations of the mean. For K = 3, at least 8/9 (or 89%) of all values lie within 3 standard deviations of the mean. Slide 13
Definition The coefficient of variation (or CV) for a set of sample or population data, expressed as a percent, describes the standard deviation relative to the mean. Sample Population s CV = 100% x CV = σ 100% µ Slide 14 7 Recap In this section we have looked at: Range Standard deviation of a sample and population Variance of a sample and population Range rule of thumb Empirical distribution Chebyshev s theorem Coefficient of variation (CV) Slide 15