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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.

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2 Answers 2

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 $0$.

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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.

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