# How do you interpret a 95% confidence interval?

Jan 19, 2017

It is an interval for which we have 95% confidence that it includes the true parameter.

#### Explanation:

Many people think that a confidence interval says something about our confidence of the location of a parameter (like $\mu$, a population mean), but this is not exactly true. What a confidence interval actually refers to is the confidence of the interval itself.

The width of a confidence interval is proportional to the estimate we get for ${\sigma}^{2}$ (the population's variance). The smaller our ${\hat{\sigma}}^{2}$, the smaller the C.I. will be. A generalized formula for a 95% C.I. for $\mu$ is:

$\hat{\mu} \pm {t}_{\frac{\alpha}{2}} \sqrt{{\hat{\sigma}}^{2} / n}$

In turn, our estimate for ${\sigma}^{2}$ depends on the data set we happened to get. Repeating the experiment will yield a new confidence interval, with a different center and a different width. Thus, it would be foolish to say that the unknown parameter has any probability of being in any one of these intervals we would be getting, due to their wandering locations and sizes.

Consider holding a single 95% C.I. fixed, and then repeating the experiment several times, getting several new $\hat{\mu}$'s. Under the false notion that a C.I. predicts the value of $\mu$ with 95% probability, we may naively expect 95% of the new $\hat{\mu}$'s to be in the first C.I. we fixed. But if that one C.I. was off to the side of the real $\mu$, and/or perhaps fairly narrow, it's not hard to imagine that a new $\hat{\mu}$ will have much less than a 95% chance of being in that first C.I. we fixed.

What we're really saying is that, any time we perform the experiment, the interval we calculate has a 95% chance of including the parameter, rather than the parameter having a 95% chance of being in any one randomly calculated interval. More specifically, if the experiment were repeated several times, then 95% of the C.I.'s obtained would cover the true parameter value.

It's not an easy concept to wrap your head around, it's true; but give it some time. Remember: it's not a 95% chance that a dart will hit the board, but rather a 95% chance that a board will fall on the dart.