Lecture Slides. Section 3-3 Measures of Variation

Transcription

1 Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola 1 Slide 1 Section 3-3 Measures of Variation Part B Created by Tom Wegleitner, Centreville, Virginia Slide 2

2 In a data list, every value falls within some number of standard deviations from the mean. For example, = 50 and s = 10, Then, any value within the interval, is within one standard deviation of the mean. Any value within the interval is within two standard deviations of the mean. 2 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 =43.2 =6.0 1 : : Slide 4

3 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) 3 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

4 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: 4 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 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) = Maximum usual value = (mean) + 2 x (standard deviation) = Interpretation: Slide 8

5 Definition Empirical ( ) 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. 5 Slide 9 The Empirical Rule Slide 10

6 The Empirical Rule 6 Slide 11 The Empirical Rule Slide 12

7 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. 7 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

8 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) 8 Slide 15