# Averaging averages and standard deviations

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

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.

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).

• Essentially you're saying that if the distribution of physics-electricity scores looks different than the distribution of physics-mechanics scores, it's not valid to just average the averages and stds? Commented Jul 13, 2015 at 17:04
• Thanks for your response. (1) So, given a sample population mean and sd and a different sample population mean and sd, there's no way to find a global population mean and sd without having access to each data point that made up those population samples? (since to do a t-test, you would need all of those points) What if I'm only given the sample mean and std? Then what? (2) My question: is the proportion of time spent on each subject predictive of the score. I therefore need to combine the subjects in such a way. Commented Jul 13, 2015 at 17:05
• (1) No, the idea is that population mean and SD are never obtainable; statistics is all about estimating these quantities to the best of our ability. The $t$-test is this: $$t = \frac{(\bar{X}_{pm} - \bar{X}_{pe}) }{ \sqrt{ \frac{S_{pm}^2}{n_{pm}} + \frac{S_{pe}^2}{n_{pe}} } }$$ in which the $n$ values are the number of data (students) used to calculate the means. You have all of these quantities. Now, large values of $t$ reject the hypothesis that $\mu_{pm} = \mu_{pe}$. Use this website to get the result of the test. Commented Jul 13, 2015 at 17:47
• (1 cont.) If the answer is that the populations aren't significantly different, the proper course would be to take the original data, use all the physics scores and get a total physics mean and standard deviation. As you don't have the original data, my recommendation is to do a weighted average of the sample means. E.g. $$\bar{X}_{weighted} = \frac{\bar{X}_{pe}*S_{pe} + \bar{X}_{pm}*S_{pm}}{S_{pm} + S_{pe}}$$ Now, this is a contentious solution but it will serve to give you the "mean" physics score using information about the quality of the averages (their SDs) that you do have. Commented Jul 13, 2015 at 18:02
• [I thought I would interrupt with my credentials: I'm almost done with a M.S. in Statistics] (2) This is the tricky part: Regression will probably give you the answer that you're looking for. In EXCEL, put the proportions of time spent on the X axis (as you've done above) and the three means on the Y axis. Excel should have a "line of best fit". This returns a slope and an intercept: The intercept is the quantity that will answer your question. So, if Excel returns $$\text{Average Grade}= 0.01 + 0.5*\text{Proportion}$$ Then you know that a 1-unit increase in proportion corresponds to... Commented Jul 13, 2015 at 18:08

I can not comment so I am answering this directly.

Your problem is to find if there is a relationship between proportion spent and average score. Let us ignore standard deviation as of now and focus on the average score. I see three problems.

1. It is risky to take averages of mechanics and electricity score - as you do not know how much proportion is spent for each of these individually. Assume proportion spent on mechanics is 0.3. It will establish a negative relationship between proportion and average score. However if proportion spent on mechanics is 0.2, it will establish a positive relationship between proportion spent and average score.
2. An exactly similar case exists for Biology score.
3. Number of data points that you have are too less to establish a relationship. And given that total Proportion spent in English is less than that spent in Biology, but average score is higher in English, the expected relationship of higher proportion implying higher average score is not valid.

May be there are more parameters which need to be considered - Difficulty of exam/subject, Interest of students, IQ level of students taking the tests etc. However, even if you do have additional parameters, with only 3-4 data points, it is hard to establish a relationship.