Transcription of Variance and standard deviation (grouped data)
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Variance and standard deviation (grouped data)
Variance and standard deviation (grouped data) IntroductionIn this leaflet we extend the definitions of Variance and standard deviation to data which has a set of values, which we denote by 2,isdefined as 2= f(x x)2nwhere xis the mean,xstands for each data value in turn, andfis the frequency with which datavalue,x,occurs. Note that f= alternative, yet equivalent formula, which is often easier to use is 2= fx2n x2where x= fxn(The Mean)Worked exampleFind an estimate of the Variance and standard deviation of the following data for the marks obtainedin a test by 88 (x)0 x<1010 x<2020 x<3030 x<4040 x<50 Frequency (f)616242517 Wecan show the calculations in a table as follows:MarksMid Intervalffxx2fx2 Value (x)0 x<1056302515010 x<2015162402253 60020 x<30252460062515 00030 x<4035258751 22530 62540 x<504517765202534 425 Total88251083 800 Mean x= fxn= mathcentre June 17, 2003 Variance 2= fx2n x2=8380088 251088 2= = (2 dp) standard deviation = an estimate of the standard deviation of the following:1.
Variance σ 2= fx2 n −x¯ 83800 88 − 2510 88 2 = 952.273−813.546 = 138.727 = 138.73 (2 dp) Standard deviation = √ 138.727 =11.78 Exercises Find …
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