Algebra for data confidence

Question

Very often, we use data which are derived from some measurements. These measurements usually have a confidence measure associated which tells how reliable the measure is, or how confident we are about the measure. For example, we often see some confidence intervals associated with various polls.

I was wondering if there is a theory or algebra about combining multiple measurements and the resulting confidence measure of the aggregate. For example, if I measure $X$ with a confidence interval $\pm x\%$, $Y$ with confidence interval $\pm y\%$, what can I say about the confidence interval of $X+Y$? Similarly, what can we say about the confidence intervals for other operators?

Is there an algebra for this?

Answer 1

[I note that there's some lack of clarity in the question; confidence intervals apply to things like parameters, as well as means or other functions of parameters; if we're talking about intervals for data that would be other kinds of interval (prediction intervals, tolerance intervals and so on). I'll proceed as if we're discussing something like means.]

If we're sticking with typical-sized polls so we have the CLT kicking in; then we're just dealing with the variances of normally distributed quantities. It depends on the dependence (specifically, the covariance) between the quantities.

$\rm{Var}(X + Y) = \rm{Var}(X) + \rm{Var}(Y) + 2 \rm{Cov}(X,Y)$

$\rm{Var}(X - Y) = \rm{Var}(X) + \rm{Var}(Y) - 2 \rm{Cov}(X,Y)$

(that doesn't rely on normality, it's general; the meaningfulness of the resulting confidence intervals depends on normality)

The width of the confidence intervals for the proportions $X$ and $Y$ and for their sum or difference are based off their respective standard errors (the square root of the variance).

If $X$ and $Y$ are independent (based on different polls for example) then the variances add because the covariances are $0$.

So square the width of the CI's for $X$ and $Y$, add them, take the square root. That's the width of the CI for the sum or difference.

If $X$ and $Y$ are two proportions from the same poll, that is wrong, since their covariance is negative. If they add to 100% or nearly so, directly add the widths of their CIs to get the width of the difference. (For the sum, the variance will be 0 - or nearly so if they don't quite add to 100% - and the width will be a multiple of the square root of that). Estimates for the covariances can actually be calculated in general, using results for the multinomial distribution.

Answer 2

I don't know if I would describe it as a special algebra per se, but the essential idea you are getting at is the Central Limit Theorem. The CLT is, in fact, one of the cornerstones of statistics. Although we usually discuss the CLT in terms of the mean, there is an obvious connection between the mean of a set of numbers and their sum. You can explore this important topic by reading the linked Wikipedia page, or by reading threads related to the topic on CV by searching on the central-limit-theorem tag. Here are a couple of good threads to get you started:

• What intuitive explanation is there for the central limit theorem?
• Understanding central limit theorem