**Property 1**: If *x̄* is the mean of the sample *S = *{*x _{1}, x_{2}, …, x_{n}*}, then the sample variance can be expressed by

it follows that

**Property 2**: If *µ* is the mean of the population *S = *{*x _{1}, x_{2}, …, x_{n}*}, then the population variance can be expressed by

Proof: Similar to the proof of Property 1.

**Property 3**: If the population {*x _{1}, x_{2}, …, x_{n}*} has mean

*µ*and standard deviation

_{x}*σ*and the population {y

_{x}_{1}, y

_{2}, …, y

*} has mean*

_{m}*µ*

_{y }and standard deviation

*σ*

_{y}, then the variance of the combined population is

Thus if *µ _{x}* =

*µ*

_{y }the combined population variance would be

**Property 4**: If the sample {*x _{1}, x_{2}, …, x_{n}*} has mean

*x̄*and standard deviation

*s*

_{x}and the sample {y

_{1}, y

_{2}, …, y

*} has mean*

_{m}*ȳ*and standard deviation

*s*

_{y}, then the variance of the combined sample is

Thus if *x̄* = ȳ, the combined sample variance would be

Proof: In general for a sample of size *k*, the sample variance can be calculated in the same way as the population variance of the same size, except that the result needs to be multiplied by *k*/(*k* – 1). Thus from the formula in Property 3 we get

from which the result follows easily.

In my opinion the formulas for property 3 to have errors in the term:

n*var(X) + m*var(Y),

whereas the term should have been:

m*var(X) + n*var(Y),

given that x has m members and y has n members.

The same error of applying the wrong membership counts to X and Y exists in property 4 as well.

James,

You are correct. X should have n elements and Y should have m elements. I have now corrected the website. Thanks very much for catching this error.

Charles

Dr. Zaiontz,

Thank you, this has helped me so much. One thing I’d like to know is how I can calculate the standard deviation of multiple samples with different ns, mus, and sigmas. Property 4 shows how you can do it with 2 samples and I’m working with 51, and I could apply Property 4 50 times to combine all 51 samples, but I was wondering if there was a less cumbersome manner of doing this.

Thanks,

Cave

Cave,

I have just added a new example (Example 5) to the Measures of Variability webpage, which shows how to perform the calculation you are looking for. I have calculated the variance, but the standard deviation is similar. Thanks for your question.

Charles