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Say a set of 5 participants completed two subtests (A and B). The scores on these subtests can be summed to get a total test score. Here is the dummy data:

Participant Subtest A Subtest B Total test
1 5 9 14
2 7 0 7
3 5 6 11
4 3 3 6
5 5 2 7

Now, say we don't have access to this raw data. We only know the summary statistics of the subtests, as follows:

  • Subtest A mean: 5.00
  • Subtest A SD: 1.41
  • Subtest B mean: 4.00
  • Subtest B SD: 3.54

If I wanted to know the total mean the for the total test, I gather I can just sum the means from the subtests (5.00 + 4.00).

But how can I calculate the total SD for the total test? I've come across one possible calculation: sqrt((1.41^2)+(3.54^2)). But when I've trialed this, I don't find that it replicates the 'true' standard deviation for the total test.

Can anyone advise? The only other advice I've found on this matter assumes that the SDs I'm trying to combine are for independent groups, rather than for repeated measures.

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  • $\begingroup$ The accepted answer to this question might help you: stats.stackexchange.com/questions/117741/…. $\endgroup$ Feb 27, 2023 at 11:25
  • $\begingroup$ Thank you! This does help - apparently "sqrt((1.41^2)+(3.54^2))" assumes that subtests A and B are completely independent, when they aren't. We'd need to build in the covariance between the two. Which opens another question - is it possible to calculate the covariance between subtests A and B using just their means, SDs and sample size? $\endgroup$
    – Alice
    Feb 27, 2023 at 11:43
  • $\begingroup$ @Alice for the covariance between A and B you need the average of A*B. $\endgroup$
    – utobi
    Feb 27, 2023 at 11:58
  • $\begingroup$ @Alice The best you could do with just the individual standard deviations is a lower or upper bound for the total standard deviation under the assumptions of perfect positive or negative correlations between the tests. Otherwise no, you need more information on how correlated the subtests are. $\endgroup$ Feb 27, 2023 at 12:30

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