Interpretation of Confidence Intervals What does a 95% confidence interval tell you? In order to answer that question, I must first review some basic definitions. Basic Definitions The following definitions are adapted from Valerie Easton and John McColl’s Statistics Glossary (http://www.stats.gla.ac.uk/steps/glossary/basic_definitions.html). Population: A population is an entire collection of people, animals, plants or things from which we may collect data. It is the entire group we are interested in, which we wish to describe or draw conclusions about.1 Sample: A sample is a group of units selected from a population. Parameter: A parameter is a value used to represent a certain population characteristic. It is usually unknown, and therefore has to be estimated. For example, the population mean ( µ ) is a parameter that is often used to indicate the average value of a quantity. Statistic: A statistic is a quantity that is calculated from a sample of data. It is used to estimate the unknown value of the corresponding population parameter. For example, the mean of the data in a sample ( X ) is used to estimate the mean in the population from which that sample was drawn ( µ ); or the mortality rate in a sample is used to estimate the true mortality rate in the population. Confidence Interval: A confidence interval gives an estimated range of values that is likely to include an unknown population parameter (e.g., a mean, a rate, an odds ratio, etc). The estimated range is calculated from a given sample. A Thought Experiment Imagine drawing all possible samples of a given size from the same population, and calculating a confidence interval for each sample.2 A certain percentage of those intervals will contain the unknown population parameter. Confidence intervals are usually calculated so that this percentage, the confidence level, is 95%.3 The width of the confidence interval gives an indication of how uncertain we are about the unknown parameter (see precision). The wider the interval is, the less certain we are. A very wide interval may indicate that more data should be collected before anything very definite can be said about the parameter. (Increasing the sample size decreases the width of the confidence interval.)

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This is the definition in statistics. Note that epidemiologists may sometimes use the word population where a statistician would use sample. 2 You would not do this in practice, of course. But thinking about it helps one to achieve a conceptual understanding of confidence intervals. 3 However, you may also encounter 90%, 99%, and 99.9% confidence intervals on occasion.

B. Weaver (27-Aug-2008)

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Interpretation of Confidence Intervals An Example This example is taken from the Wiki Books Introduction to Medical Statistics (http://en.wikibooks.org/wiki/Introduction_to_Medical_Statistics). It concerns a hypothetical population of (human) body temperatures, which has a mean of 37ºC with a standard deviation (SD) of 0.407. Figure 1 shows the means and 95% confidence intervals from 50 random samples (all of the same size) drawn from that population. Ninety-five percent of 50 is 47.5. That is the number of confidence intervals (out of 50) expected to contain the true population parameter, 37ºC. But of course, we cannot have 47.5 intervals—the actual number must be a whole number. In fact, 48 of the 50 confidence intervals contain the parameter. The only two confidence intervals that do not contain 37ºC are from samples 20 and 40 (indicated with arrows).

Figure 1. Fifty 95% confidence intervals based on 50 samples from a hypothetical (normally distributed) population of body temperatures with a mean of 37°C and SD of 0.407.

B. Weaver (27-Aug-2008)

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