Changing the constant (i.e., the "1.96") changes the percentage associated with the confidence interval; for example, a 90 percent confidence interval for a mean created from a simple random sample is:

Confidence intervals can be created for other statistics as well, but creating those confidence intervals can be more difficult [1].

The form given above for a confidence interval is not complicated in itself; what makes confidence intervals complicated to interpret are two factors:

1. The assumptions that must be valid in order for the confidence interval to be valid; and

2. Understanding the proper interpretation of a confidence interval.

There are two basic assumptions that must be true for a confidence interval to "work." First, the data used to develop the statistic for which the confidence interval is created must be mostly "symmetric," meaning, they can't have too much skew. Second, there must be enough data used to develop the statistic for which the confidence interval is created. Usually, there should be at least 30 data points available. These two assumptions interact when we are creating a confidence interval for a mean, in that the more data that are collected/available, the more skew we can tolerate in those data.

If these assumptions are not valid, meaning, if the data have too much skew or there aren't enough data, then confidence intervals derived using the equations above may not be meaningful.

Interpreting the meaning of a confidence interval can be tricky. What is random about a confidence interval is the endpoints of that interval.

For example, let's assume that we are to take a simple random sample of 100 members of a population for the purpose of calculating the mean height (in centimeters) of members of that population. Imagine that we could repeat our experiment 100 times, that is, collect 100 samples and, for each sample, calculate a mean and a standard error. We would be able to form a confidence interval for each of our 100 samples.

The 95 percent confidence interval is then defined as the interval such that in approximately 95 of the experiments, the parameter (true mean height) we are estimating is “captured” within the confidence interval, and in approximately five of the experiments, the parameter (true mean height) we are estimating is not “captured” within the confidence interval. Of course, we don’t repeat our experiment 100 times; we just do it once. What we are counting on is that the experiment we do is not one of the approximately five for which the confidence interval does not capture the parameter.

The illustration below demonstrates the results of repeating this experiment 50 times. The mu in the graphic represents the parameter we are estimating, that is, the true mean height of member of the population of interest. Each line represents one experiment, that is, the confidence interval calculated from one simple random sample of 100 members of the population. Note that in three cases, the 95 percent confidence interval does not contain mu, which is about what we would expect after 50 repeats of the experiment (about 5 percent of the cases).

Source: Wikipedia [disclaimer]

1. For example, the confidence interval for a median is calculated as given by www.umanitoba.ca/centres/mchp/concept/dict/ (26 December 2006), and the confidence interval for a ratio of means is calculated as given by www.graphpad.com/FAQ/images/Ci of quotient.pdf (26 December 2006).