Will the standard deviation of all data in three unequally-sized groups each with different means be

excluderho

excluderho

Answered question

2022-06-07

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.

Answer & Explanation

Ethen Valentine

Ethen Valentine

Beginner2022-06-08Added 15 answers

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.
Kiana Dodson

Kiana Dodson

Beginner2022-06-09Added 5 answers

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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