# Confidence Intervals of the Sum

I am regularly use confidence intervals (CI) of the mean and more than comfortable with using them. However, in one particular analysis I am more interested in CI of the sum rather than the CI of the mean.

For example, let's say we have a dataset of elements between 1 and 9, where n=100. This gives a mean of 5.3 and a sum of 530. Is it possible to identify the CI for the sum?

The assumption in this question is that the sum is not fixed and, hence, has some degree of variance. So far my search has come up with nothing. The best approach seems to use a bootstrapping method.

• Usually you construct confidence intervals for some parameter. What is the parameter underlying the sum? A population total or something else? – kjetil b halvorsen Sep 26 '17 at 9:02

The CI for the sum is equal to the CI of the mean, multiplied by the number of observations. So for example, if your CI for your mean of 5.3 was [4.8, 5.8] then the CI for the sum is [480,580].

That was a simple answer but there are a couple of points to bear in mind:

• Arguably, the concept of "CI for the sum" doesn't make sense. We actually observed the sum, and it was definitely 530. There's no argument with that (unless there was measurement error). What we are really saying is "if I took the sum of another 100 independent observations generated in a similar way, what range would I expect that sum to lie in 95% of the time".

• If you have a large enough sample, the CI for the mean will become very small and ultimately shrink towards a point. This will probably not happen for the sum.

• Thanks for the explanation and clarification. I am happy with that. – Murray B Sep 26 '17 at 9:06
• I think you should multiply by the size of the population, not with the number of observations. Moreover, you should take into account a finite population correction. see e.g. stats.stackexchange.com/questions/167972/… – user83346 Sep 26 '17 at 9:08
• @fcop I guess that depends if you interpret the "sum" as referring to the sample or the entire population. I took the former interpretation, you have taken the latter. I guess both are legitimate! – JDL Sep 26 '17 at 9:10
• @Murray B: thanks! If you'd be kind enough to accept the answer it'd be much appreciated. – JDL Sep 26 '17 at 9:11
• Well the sum of the sample CI sounds strange to me: if I look for a 95% CI for the mean I am finding a random interval that has a change of 0.95 to contain the 'true' population mean. If I look for a 95% CI for the (population) sum I look for a random interval that has 0.95 chance of containing the population total, what would be the interpretation of a 95% CI for the sample sum ? – user83346 Sep 26 '17 at 9:49

A confidence interval is a range within which a specified % of the samples fall.

So CI of 95% means that 95% of the values in the sample fall inside that specific range.

Hence asking for a 95% Confidence Interval of the sum, which contains 100% of the sample, doesn't make sense.