Measures of Dispersion Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. HAWKES LEARNING SYSTEMS math courseware specialists Section 3.2 Measures of Dispersion

A measure of dispersion tells how spread out the data values are. HAWKES LEARNING SYSTEMS math courseware specialists Numerical Descriptions of Data 3.2 Measures of Dispersion A measure of dispersion tells how spread out the data values are. The measures of dispersion we will calculate are the range, standard deviation, and variance.

The range is the simplest measure of dispersion. HAWKES LEARNING SYSTEMS math courseware specialists Numerical Descriptions of Data 3.2 Measures of Dispersion Calculating the range: The range is the simplest measure of dispersion. Range = Maximum - Minimum Calculate the range of the following data sets: 2 2 2 3 5 6 9 15 2 3 5 6 7 8 Solution: 15 - 2 = 13 Solution: 8 - 2 = 6

The variance is the average squared deviation from the mean. HAWKES LEARNING SYSTEMS math courseware specialists Numerical Descriptions of Data 3.2 Measures of Dispersion Calculating the variance: The variance is the average squared deviation from the mean. Formula:

HAWKES LEARNING SYSTEMS math courseware specialists Numerical Descriptions of Data 3.2 Measures of Dispersion Calculate the variance of the following sample data: 4.2 5.3 2.9 6.7 1.5 Solution: First calculate the mean of the data. The mean is 4.12. Sample Variance xi 4.2 4.2 – 4.12 = 0.08 0.0064 5.3 5.3 – 4.12 = 1.18 1.3924 2.9 2.9 – 4.12 = –1.22 1.4884 6.7 6.7 – 4.12 = 2.58 6.6564 1.5 1.5 – 4.12 = –2.62 6.8644

Therefore the sample variance is HAWKES LEARNING SYSTEMS math courseware specialists Numerical Descriptions of Data 3.2 Measures of Dispersion Solution (continued): Therefore the sample variance is When calculating the variance, round to one more decimal place than what is given in the data.

HAWKES LEARNING SYSTEMS math courseware specialists Numerical Descriptions of Data 3.2 Measures of Dispersion Calculating the standard deviation: The standard deviation tells the average distance the data values lie from the mean. Formula:

HAWKES LEARNING SYSTEMS math courseware specialists Numerical Descriptions of Data 3.2 Measures of Dispersion Calculate the standard deviation of the following sample data: 3.1 4.6 2.9 5.1 2.2 4.8 3.3 Average = 3.714 s 2 = [(3.1 – 3.714) 2 + (4.6 – 3.714) 2 + (2.9 – 3.714) 2 + (5.1 – 3.714) 2 + (2.2 – 3.714) 2 + (4.8 – 3.714) 2 + (3.3 – 3.714) 2 ] ÷ (7 -1) s 2 = 7.3886 ÷ 6 = 1.2314 s = √ 1.2314 = 1.1097

Mean = AVERAGE() Sample Standard Deviation = STDEV() Sample Variance = VAR() Population Standard Deviation = STDEVP() Population Variance = VARP()

Used when a set of data is approximately bell-shaped. HAWKES LEARNING SYSTEMS math courseware specialists Numerical Descriptions of Data 3.2 Measures of Dispersion Empirical Rule: Used when a set of data is approximately bell-shaped. Approximately 68% of the data lies within 1 standard deviation of the mean. Approximately 95% of the data lies within 2 standard deviations of the mean. Approximately 99.7% of the data lies within 3 standard deviations of the mean. Z1 = 2183 – 3325 = – 1142/571= –2 Z2 = 4467 – 3325 = 1142/571= 2 P(2183 < x < 4467) = 95%

HAWKES LEARNING SYSTEMS math courseware specialists Numerical Descriptions of Data 3.2 Measures of Dispersion Chebyshev’s Thoerem: When the empirical rule cannot be used, Chebyshev’s Theorem can help to give a minimum estimate. The proportion of data that lie within K standard deviations of the mean is at least , for K > 1. K = 2: At least of the data lies within two standard deviations of the mean. K = 3: At least of the data lies within three standard deviations of the mean.

Copy Data from Hawkes to Excel Paste in cell A1

Copy Data from Hawkes to Excel Paste in cell A1

0.95 Z1 = (6.41 – 7.25) ÷ 0.42 = -2 Z2 = (8.09 – 7.25) ÷ 0.42 = 2 Area (-2 < z < 2) = 0.95

0.95 0.025 Z2 = (3.41 – 2.61) ÷ 0.4 = 2 Area (Z2 ≤ 2) = 0.95 + 0.025 = 0.975

Coefficient of Variation (CV) is a measure of spread Coefficient of Variation (CV) is a measure of spread. It is the Standard Deviation divided by the Mean. The higher the CV, the higher the spread.

Copy Data from Hawkes to Excel Paste in cell A1 Data set A has the largest spread