# If three groups of unequal sizes have the same standard deviation but different means, will the SD of the pooled data be the same?

Will the standard deviation of all data in three unequally-sized groups each with different means be the same if each group has the same standard deviation?

For example, I have a data set as follows:

• Group 1, Makes up 30% of individuals, Mean = 12, SD = 3
• Group 2, Makes up 50% of individuals, Mean = 17, SD = 3
• Group 3, Makes up 20% of individuals, Mean = 11, SD = 3

I've been asked to calculate whether the SD for all individuals together is equal to 3, or larger than 3. My instinct is to say that it will be larger than 3 given that the the means are quite far apart, but can I be certain given just the information provided...? Any help here would be much appreciated.

The standard deviation, as you surmise, will be larger than $3$. The issue is the different means.

The fact becomes more clear if we imagine the individual standard deviations to be very small, say, $0.1$, or, even worse, $0$. The standard deviation of the pooled data is obviously not close to $0$.

• Thanks Andre - that does indeed make things clearer; I was wracking my brains to find a distribution of data that could act as a counterexample to the hypothesis that the SD is greater, but couldn't find one. Your answer helps to explain why. – Mr.A Mar 13 '13 at 21:42

As André said, the better counter-example is:

• Group 1, Makes up 30% of individuals, Mean = 12, SD = 0
• Group 2, Makes up 50% of individuals, Mean = 17, SD = 0
• Group 3, Makes up 20% of individuals, Mean = 11, SD = 0

As with these data it appears clear that the overall standard deviation will be larger than the standard deviations of the three groups.