In statistics, a prediction interval bears the same relationship to a future observation that a confidence interval bears to an unobservable population parameter. Prediction intervals predict the distribution of invidividual points, whereas confidence intervals estimate the true population mean or other quantity of interest that cannot be observed.

Example

Suppose one has drawn a sample from a normally distributed population. The mean and standard deviation of the population are unknown except insofar as they can be estimated based on the sample. It is desired to predict the next observation. Let n be the sample size; let μ and σ be respectively the unobservable mean and standard deviation of the population. Let X₁, ..., X_n, be the sample; let X_n+1 be the future observation to be predicted. Let

$\backslash overline\{X\}\_n=(X\_1+\backslash cdots+X\_n)/n$

and

$S\_n^2=\{1\; \backslash over\; n-1\}\backslash sum\_\{i=1\}^n\; (X\_i-\backslash overline\{X\}\_n)^2.$

Then it is fairly routine to show that

$\{X\_\{n+1\}-\backslash overline\{X\}\_n\; \backslash over\; \backslash sqrt\{S\_n^2+S\_n^2/n\}\}\; =\; \{X\_\{n+1\}-\backslash overline\{X\}\_n\; \backslash over\; S\_n\backslash sqrt\{1+1/n\}\}$

has a Student's t-distribution with n − 1 degrees of freedom. Consequently we have

$P\backslash left(\backslash overline\{X\}\_n-A\; S\_n\backslash sqrt\{1+(1/n)\}\backslash leq\; X\_\{n+1\}\; \backslash leq\backslash overline\{X\}\_n+A\; S\_n\backslash sqrt\{1+(1/n)\}\backslash ,\backslash right)=p$

where A is the 100(1 − (p/2))^th percentile of Student's t-distribution with n − 1 degrees of freedom. Therefore the numbers

$\backslash overline\{X\}\_n\backslash pm\; A\; \{S\}\_n\backslash sqrt\{1+(1/n)\}$

are the endpoints of a 100p% prediction interval for X_n + 1.

See also

• Confidence interval
• Extrapolation
• Prediction
• Regression analysis
• Seymour Geisser
• Trend estimation

References

• Chatfield, C. (1993) "Calculating Interval Forecasts," Journal of Business and Economic Statistics, 11 121-135.
• Meade, N. and T. Islam (1995) "Prediction Intervals for Growth Curve Forecasts," Journal of Forecasting, 14 413-430.

Statistics