# Confidence interval for sum of means

I have a question about how to calculate the confidence interval for the sum of means from different samples.

For example, I take three random samples of students from a school and give the first sample a math test (sample size $n_1 = 50$), the second sample a reading test ($n_2 = 40$), and the third sample a physics test ($n_3 = 30$).

Let us say the students from the three samples are mutually exclusive. How can I calculate the confidence interval for the sum of mean scores, i.e., mean_math + mean_reading + mean_writing? Is there a command in Stata, SAS, or Matlab that I can use? Thanks.

• Why would you need to sum the means? – Ayalew A. Jan 29 '15 at 7:39
• To construct the CI for the total score. – frostman Jan 29 '15 at 7:42
• All the three tests possibly have different scales, different means and variances - summing them does not sound like a good idea. – Tim Jan 29 '15 at 7:58
• For three different (independent) samples each taking a different test, I don't think a total score would give any sense. – Ayalew A. Jan 29 '15 at 8:01
• .......Even just for maths test, would the total test score of all the 50 students be meaningful? – Ayalew A. Jan 29 '15 at 8:39

The for addition/subtraction uncertainties add in quadrature. Therefore the combined confidence interval is: $$u_c = \sqrt{u_m^2+u_r^2+u_w^2}$$

For an individual test the confidence interval can be given by the standard deviation of the mean.

$$s_m = \frac{s}{\sqrt{n}}$$

where $s$ is the standard deviation of the sample. This should then be multiplied by the coverage factor to get the desired confidence interval.

$$u=ks_m$$

For a normal distribution at 95% confidence $k=1.96$ ($\simeq2$).

In matlab the easiest way to do it is to just calculate standard deviation (std()) or variance and go from there. I don't use Stata/SAS so don't know if they have any specific functions.

I would also think carefully about the meaning of your test. You should probably normalise the scores so they are all on the same scale. Additionally I suspect they may be some correlation between results for a single student taking all three tests. That is students are likely to good/bad in all the tests. If this is the case you would get a larger confidence interval if you take a sample of different students total score than for the total of different samples on each test (what you currently do).

(Aside on the general case)

In general for any function $f(x_1, x_2, ... , x_m)$ where all $x_i$ are independent of each other and have associated uncertainty, $u_i$, the combined uncertainty $u_c$ can be given by: $$u_c^2 = \sum_{i=1}^m\left(\frac{\partial f}{\partial x_i}^2\right)u_i^2$$ You should make sure all $u_i$ have the same coverage factor. You can calculate uncertainty $k=1$ and then expand the combined uncertainty if you need to.