Variance and standard deviation (grouped data)

1 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.

2 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. a)LifetimeFrequencyb)IncomeFrequencyc)Ho useFrequency(hours) of(1000 s )pricescomponents(1000 s )300 l<4001310 i<15940 p<605400 l<5002515 i<201660 p<809500 l<6006620 i<252280 p<10015600 l<7005825 i<308100 p<1208700 l<8003830 i<355120 p<14032.

3 The grouped frequency table shows the length of service in years of employees who have beenworking for a company for at least ten an estimate of the standard deviation of the length of service of these of Service (x)10 x<1515 x<2020 x<2525 x<3030 x<4040 x<50 Frequency (f)3042231384 Answers1. a) hoursb) 5617c) 21 9942. years (2 dp)Note that the units of standard deviation are the same as the original data units. This is not trueof the Variance where the units would be the square of the original data mathcentre June 17, 2003