Expert: Abe Mantell - 1/3/2007

Question
I am trying to calculate the new standard deviation when data from two groups is combined.
Group 1: N=22, mean = 8.36, SD = 6.93
Group 2: N = 29, mean = 8.28, SD = 7.33
Group 1 & Group 2: SD = ?
I do not have data for the individuals, only these summary statistics. I would really like to know the equations/process for calculating this as I need to do this again in the future. Any help you could offer would be greatly appreciated, this has baffled me for months now :(


Answer
Hello Hunna,

I will assume the SD's you gave are "sample SD's"...OK?
So that the sample SD formula is:
SD=sqrt[sum(mean-x)^2/(n-1)], or we can use
SD=sqrt[(n*sum(x^2)-(sum(x))^2)/(n(n-1))]
Using this, we obtain for Sample I:
Since the mean is 8.36 with n=22
...this gives sum(x)=8.36*22=183.82
now we can solve for sum(x^2)...getting
sum(x^2)=[(6.93^2)*22*21+183.82^2]/22=2544.42255455
Now do the same thing for Sample II:
sum(x)=8.28*29=240.12, and
sum(x^2)=[(7.33^2*29*28+240.12^2]/29=3492.6028

So, for the combined data, we get:
sum(x)=183.82+240.12=423.94, and
sum(x^2)=2544.42255455+3492.6028=6037.02535455

Now use the formula fod SD (with n=22+29=51, etc.)
SD=sqrt[(51*6037.02535455-423.94^2)/(51*50)]
= 7.0894

OK???

Abe