Section32MeasuresofDispersion32部分分散的措施

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,*,Section 3.2 Measures of Dispersion,Range,Variance,Standard deviation,Empirical Rule for bell shaped distributions,Chebyshevs,Inequality for any distribution,3-,1,Range,The,range,of a set of data is the difference between the maximum value and the minimum value.,Range=(maximum value)(minimum value),EXAMPLE,The following data represent the travel times(in minutes)to work for all seven employees of a start-up web development company.,23,36,23,18,5,26,43,Find the range.,Range=43 5,=38 minutes,The,population variance,is the sum of squared deviations about the population mean divided by the number of observations in the population,N,.,That is it is the mean of the sum of the squared deviations about the population mean.,3-,4,Variance,The,population variance,is symbolically represented by,2,(lower case Greek sigma squared).,3-,5,EXAMPLE,Population Variance,The following data represent the travel times(in minutes)to work for all seven employees of a start-up web development company.,23,36,23,18,5,26,43,Compute the population variance of this data.Recall that,3-,6,x,i,x,i,(,x,i,),2,23,24.85714,-1.85714,3.44898,36,24.85714,11.14286,124.1633,23,24.85714,-1.85714,3.44898,18,24.85714,-6.85714,47.02041,5,24.85714,-19.8571,394.3061,26,24.85714,1.142857,1.306122,43,24.85714,18.14286,329.1633,902.8571,minutes,2,3-,7,The,sample variance,is computed by determining the sum of squared deviations about the sample mean and then dividing this result by,n,1.,3-,8,EXAMPLE,Sample Variance,For the travel time data assume we obtained the following simple random sample:5,36,26.,Compute the sample variance travel time.,Travel Time,x,i,Sample Mean,Deviation about the Mean,Squared Deviations about the Mean,5,22.333,5 22.333,=-17.333,(-17.333),2,=300.432889,36,22.333,13.667,186.786889,26,22.333,3.667,13.446889,square minutes,3-,9,Standard Deviation,The,standard deviation,of a set of sample values is a measure of variation of values about the mean.,Population standard deviation:,=square root of the population variance,Sample standard deviation:,s,=square root of the sample variance,so that,3-,11,EXAMPLE,Population Standard Deviation,The following data represent the travel times(in minutes)to work for all seven employees of a start-up web development company.,23,36,23,18,5,26,43,Compute the population standard deviation of this data.,Recall,from the last objective that,2,=129.0 minutes,2,.,Therefore,3-,12,EXAMPLE,Sample Standard Deviation,Recall the sample data 5,26,36 results in a sample variance of,square minutes,Use this result to determine the sample standard deviation.,3-,13,1.500.791.011.660.940.67,2.531.201.460.890.950.90,1.882.941.401.331.200.84,3.991.901.001.540.990.35,0.901.230.921.091.722.00,3.500.000.380.431.823.04,0.000.260.140.602.332.54,1.970.712.224.540.800.50,0.000.280.441.380.921.17,3.082.750.363.102.190.23,Wait Time at Wendys,Wait Time at McDonalds,3-,14,EXAMPLE,Comparing Standard Deviations,EXAMPLE,Comparing Standard Deviations,Determine the standard deviation waiting time for Wendys and McDonalds.,Which is the better company in terms of waiting times?,3-,15,EXAMPLE,Comparing Standard Deviations,Determine the standard deviation waiting time for Wendys and McDonalds.,Sample standard deviation for Wendys:,0.738 minutes,Sample standard deviation for McDonalds:,1.265 minutes,3-,16,For many observations,especially if their histogram is bell-shaped,Roughly,68%,of the observations in the list lie within,1 standard deviation,from the average,And,95%,of the observations lie within,2 standard deviations,from the average,Average,Ave-s.d.,Ave+s.d.,68%,95%,Ave-2s.d.,Ave+2s.d.,The empirical rule for bell shaped distributions,3-,18,The Empirical Rule,The Empirical Rule,The Empirical Rule,EXAMPLE,Using the Empirical Rule,The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor.,414843383537444444,627577588239855554,676969706572747474,606060616263646464,545455565656575859,454747484850525253,3-,22,Compute the population mean and standard deviation.,(b)Draw a histogram to verify the data is bell-shaped.,(c),Determine the percentage of patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule.,(d)Determine the percentage of patients that have serum HDL between 34 and 69.1 according to the Empirical Rule.,(e)Determine the actual percentage of patients that have serum HDL between 34 and 69.1,(use the raw data directly,not the empirical rule for this question.See how close the empirical rule approximation was!),3-,23,(a)Using a TI-83 plus graphing calculator or Excel,we find,(b),3-,24,22.3 34.0 45.7 57.4 69.1 80.8 92.5,(e)45 out of the 54 or 83.3%of the patients have a serum HDL between 34.0 and 69.1.,(c)According to the Empirical Rule,99.7%of the,patients that have serum HDL within 3 standard deviations of the mean.,(d)13.5%+34%+34%=81.5%of