Solved – Averaging averages and standard deviations

meanstandard deviation

I am comparing exam scores for different subjects: physics, biology, english. I am looking at what proportion of the curriculum is spent on each subject. The number of students is fixed: the same for all classes. 

physics mechanics score   av = 10; sd = 2 
physics electricity score av = 15; sd = 5
biology score             av = 10; sd = 3
english score             av = 12; sd = 6

physics proportion        av = 0.5
biology A proportion      av = 0.1
biology B proportion      av = 0.2
english proportion        av = 0.2

Obviously I can say that english was taught 0.2 of the time and has a score of 12 with sd 6. However, biology and physics aren't so straightforward.

Is the following reasonable:

For physics, I average the scores (again, same number of students): to get

physics = 12.5. To get the standard deviation, I

sqrt(2*2+5*5)=5.4.

Therefore my physics score is 12.5 with std 5.4, and I compare that with the physics proportion, 0.5.

Lastly, for biology, I know my score is 12 and my standard deviation is 6. To see what proportion of the time is spent on biology, I simply add: 0.2 + 0.1 = 0.3.

I therefore have:

physics (0.5) –> 12.5, 5.4
biology (0.3) –> 12, 6
english (0.2) –> 12, 6

Plot of time spent v scores

I am wondering if my method for combining the averages and standard deviations (essentially, averaging them) for physics and for combing the proportions (adding them) for biology were reasonable for this problem specifically.

Best Answer

I don't think that this is valid. What you have are:

  • $\bar{X}_{pm} = 10$, the sample mean of the Physics-mechanics scores,
  • $S_{pm} = 2$, the sample standard deviation of the Physics-mechanics scores
  • $\bar{X}_{pe} =15$, the sample mean of the Physics-electricity scores
  • $S_{pe} = 5$, the sample standard deviation of the Physics-electricity scores
  • $\bar{X}_{b} = 10$, the sample mean of the biology scores
  • $S_{b} = 3$, the sample standard deviation of the biology scores
  • $\bar{X}_{e} = 12$, the sample mean of English scores, and
  • $S_{e} = 6$, the sample standard deviation of English scores.

First of all, I can't imagine that all of these scores are sampled from the same population, so combining them in the way you did is a bit odd. What I mean by that is that the Physics-mechanics scores come from some distribution, $f_{pm}$, ostensibly with population mean $\mu_{pm}$ and population standard deviation $\sigma_{pm}$. The sample mean and sample standard deviation are estimators of the respective population quantities. The Physics-electricity scores come from some presumably other distribution, $f_{pe}$, with population mean $\mu_{pe}$ and population standard deviation $\sigma_{pe}$. The biology scores and English scores would be samples from two other distributions.

By "averaging" all the sample means, the assumption that you're making is that all of the data in question are being sampled from the same population, which is dubious. At best, you could two a two-proportion $t$-test to see if the physics results come from the same distribution, and if there is statistically-significant evidence that they do, you could combine all the physics scores to get $\bar{X}_{physics}$ and $S_{physics}$.

Lastly, as for the proportions of time, what question are you trying to answer? If you want to know if the average score is higher in one discipline or another, once again the 2-mean $t$-test is the way to go. If you want to compare all three disciplines at the same time, you're looking at MANOVA (multiple analysis of variance).

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