Confidence Interval

A range of values, calculated from sample data, expected to contain the true population parameter with a specified level of confidence.

Source: Neyman 1937

A confidence interval is a range, computed from sample data, that is expected to contain the true population value with a stated level of confidence, such as ninety-five per cent. It conveys the precision of an estimate: a narrow interval means the data pin the value down tightly, while a wide one signals considerable uncertainty. By showing the span of values compatible with the data, it says far more than a single point estimate about how firmly a result is known. Expressing the uncertainty around an estimate is its purpose. Kirkwood and Sterne (2003) describe this.

Source: Kirkwood & Sterne 2003

A confidence interval is interpreted in terms of the procedure: a ninety-five per cent confidence interval means that if the study were repeated many times, about ninety-five per cent of the intervals so constructed would contain the true parameter. It is not correct, in the frequentist framework, to say there is a ninety-five per cent probability that the particular interval contains the parameter, since the parameter is fixed. So a confidence interval is interpreted as arising from a procedure that captures the true value a specified proportion of the time, conveying the precision of the estimate, with a wider interval indicating greater uncertainty and a narrower one greater precision.

Source: Neyman 1937

A confidence interval is calculated from the point estimate, its standard error, and a multiplier from the relevant distribution corresponding to the confidence level, typically as the estimate plus and minus the multiplier times the standard error. For a ninety-five per cent interval based on a normal approximation, the multiplier is about 1.96. So a confidence interval is calculated by combining the estimate with a margin based on its standard error and the confidence level, which widens with greater variability or smaller samples, and the exact method depends on the parameter and the distribution, with approximate normal-based intervals common in large samples.

Source: Neyman 1937

The width of a confidence interval is affected by the sample size, the variability in the data, and the confidence level: larger samples and lower variability give narrower intervals, reflecting greater precision, while a higher confidence level, such as ninety-nine rather than ninety-five per cent, gives a wider interval. So the width of a confidence interval reflects the precision of the estimate, narrowing with more data and less variability and widening with a higher confidence level, which means a narrow interval indicates a precise estimate and a wide one considerable uncertainty, and the interplay of these factors is considered when designing studies to achieve adequate precision.

Source: Neyman 1937

Confidence intervals are useful because they convey the precision of an estimate and a range of plausible values for the parameter, giving more information than a point estimate or a p-value alone, and helping to judge both statistical and practical significance. A wide interval warns of imprecision, and whether the interval includes a value of no effect is informative. So confidence intervals are useful for interpreting results, since they show how precisely a quantity is estimated and which values are compatible with the data, which supports better judgement than significance testing alone about the size and certainty of an effect.

Source: Neyman 1937