Page 112 - Demo

72 Chapter 1 %u2022 Analyzing One-Variable Data More important than the details of calculating sx are the properties of the standard deviation as a measure of variability: %u2022 sx is always greater than or equal to 0. sx = 0 only when there is no variability%u2014that is, when all values in a distribution are the same. %u2022 Greater variation from the mean results in larger values of sx . For instance, the widths of 70-mm strips of film produced by Machine A have a standard deviation of 0.110 mm, whereas the widths of 70-mm strips of film produced by Machine B have a standard deviation of about 0.167 mm. That%u2019s about 52% more variability in the widths of film strips produced by Machine B! 69.8 69.9 70.0 70.1 70.2Film strip width (mm)Machine AB %u2022 sx is not resistant. The use of squared deviations makes sx even more sensitive than x to extreme values in a distribution. In the earlier example, the standard deviation of the travel time to school for the 20 randomly selected U.S. high school students SOLUTION: = + + + + + + + + = = 1 3444 5 7 8 99459x 5 petsValue xi Deviation from mean x x i %u2212 Squared deviation ( ) x x i %u2212 2 1 1%u2013 5 %u20134 = ( 4%u2212 = ) 16 2 3 3 %u2013 5 %u20132 = ( 2%u2212 = ) 4 2 4 4 %u2013 5 %u20131 = ( 1%u2212 = ) 1 2 4 4 %u2013 5 %u20131 = ( 1%u2212 = ) 1 2 4 4 %u2013 5 %u20131 = ( 1%u2212 = ) 1 2 5 5 %u2013 5 0 = 0 0 =2 7 7 %u2013 5 2 = 2 4 =2 8 8 %u2013 5 3 = 3 9 =2 9 9 %u2013 5 4 = 4 1 = 6 2Sum 52 == %u2212 = 529 16.5 2sxsx = = 6.5 2.55 petsInterpretation: The number of pets these children have typically varies from the mean by about 2.55 pets. 1. Find the mean of the distribution. 3. Square each deviation. 4. Add all the squared deviations and divide by n %u20131. This gives the sample variance . 2. Calculate the deviation of each value from the mean: deviation v = alue %u2013 mean %u2211 = %u2212%u2212( )12s x xn xiFOR PRACTICE TRY EXERCISE 13. 5. Take the square root to return to the original units (pets). %u00a9 Bedford, Freeman & Worth Publishers. For review purposes only. Do not distribute.