Confidence Interval

A confidence interval is an interval that will contain a population parameter a specified proportion of the time. The confidence interval can take any number of probabilities, with the most common being 95% or 99%.

A confidence interval is the probability that a value will fall between an upper and lower bound of a probability distribution. For example, given a 99% confidence interval, stock XYZ's return will fall between -6.7% and +8.3% over the next year. In layman's terms, you are 99% confident that the returns of holding XYZ stock over the next year will fall between -6.7% and +8.3%.

Statisticians use confidence intervals to measure uncertainty. A higher probability associated with the confidence interval means that there is a greater degree of certainty that the parameter falls within the bounds of the interval. Therefore, a higher confidence level indicates that the parameters must be broader to ensure that level of confidence.

For example, suppose a group of researchers is studying the heights of high school basketball players. The researchers take a random sample from the population and establish a mean height of 74 inches. The mean of 74 is a point estimate of the population mean. A point estimate by itself is of limited usefulness because it does not reveal the uncertainty associated with the estimate; you do not have a good sense of how far this sample mean may be from the population mean. What they are missing at this point is the degree of uncertainty in this single sample.

Confidence intervals provide more information than point estimates. By establishing a 95% confidence interval using the sample's mean and standard deviation, and assuming a normal distribution as represented by the bell curve, the researchers arrive at an upper and lower bound that contains the true mean 95% of the time. Assume the interval is 72 inches to 76 inches. If the researchers take 100 random samples from the population of high school basketball players as a whole, the mean should fall between 72 and 76 inches in 95 of those samples.

If the researchers want even greater confidence, they can expand the interval to 99% confidence. Doing so invariably creates a broader range, as it makes room for a greater number of sample means. If they establish the 99% confidence interval as 70 inches to 78 inches, they can expect 99 of 100 samples evaluated to contain a mean value between these numbers.

The biggest misconception regarding confidence intervals is that they represent the percentage of data from a given sample that falls between the upper and lower bounds. For example, one might erroneously interpret the aforementioned 99% confidence interval of 70 to 78 inches as indicating that 99% of the data in a random sample falls between these numbers. This is incorrect, though a separate method of statistical analysis exists to make such a determination. Doing so involves identifying the sample's mean and standard deviation and plotting these figures on a bell curve. The confidence level describes the uncertainty associated with a sampling method. A 90% confidence level means that we would expect 90% of the interval estimates to include the population parameter. Likewise, a 99% confidence level means that 95% of the intervals would include the parameter.