Statistic (How to calculate Variance & Standard Deviation)?

Question

Frequency distribution table (in $'000)
Lower LimitUpper LimitCountCumulative Count%Cumulative %
2030337.50 7.50
31406915.00 22.50
415081720.00 42.50
516072417.50 60.00
617073117.50 77.50
718053612.50 90.00
81901372.50 92.50
911003407.50 100.00

Can I know how to compute the variace and standard deviation from the above data? What is the formula? Thanks in advance.

Answer 1

range, n, N, x,...ne,ne^2
20, 30, 3, 3,-3,. -9, 27
31, 40, 6, 9,-2,-12, 24
41, 50,8,17,-1,. -8,.. 8
51, 60,7,24, 0,.....,.....
61, 70,7,31, 1,.. +7,. 7
71, 80,5,36, 2, +10, 20
81, 90,1,37, 3,.. +3,.. 9
91,100,3,40, 4, +12, 48,
...........................3, 143
interval = 10, midpoint = 5
m = (55+10(3/40))*1,000 = 55,750
var = (100(143/40))*1,000,000 = 357,500,000
std. dev.. = 1,000sqrt(1430/4) =18,908


Strictly speaking,
m = 1,000*(3*25 + 6*35 + 8*45 + 7*55 + 7*65 + 5*75 + 1*85 + 3*95) / 40 = 55,750
var = 1,000,000*(3*30.75^2 + 6*20.75^2 + 8*10.75^2 + 7*0.75^2 + 7*9.25^2 + 5*19.25^2 + 1*29.25^2 + 3*39.25^2) / 40 = 356,937,500
sigma = 18,893

The errors in variance and std. dev. result from my failure to account for the 3/40 difference in the scaled mean.