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Prediction interval

A prediction interval bounds where a single new observation will likely fall. It is wider than the confidence interval for the mean because it must also account for the natural variability of individual observations around the fitted line.

ByHoang TruongUpdated

FrameworkRegression inference

See it move

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A regression's confidence interval bounds where the true average response lies at a given predictor value, capturing only estimation error. A prediction interval instead bounds where the next individual observation will fall, so it must also capture the natural scatter of observations around the fitted line. That extra scatter never disappears, so a prediction interval is always wider than the matching confidence interval.

Where it fits
SubjectData Analysis & StatisticsAdvancedTopicEstimation & Sampling DistributionsAdvancedTopicSimple Linear Regression & OLSAdvanced

The formula

LaTeX
PI=y^±t⋅s1+1n+(x∗−xˉ)2∑i=1n(xi−xˉ)2\text{PI} = \hat{y} \pm t \cdot s\sqrt{1 + \frac{1}{n} + \frac{(x^{*} - \bar{x})^2}{\sum_{i=1}^{n}(x_i - \bar{x})^2}}

Variables

Fitted value at the new predictor value x*
Critical t-value with n − 2 degrees of freedom at the chosen confidence level
Residual standard error of the regression
Number of observations in the sample
New predictor value at which the forecast is made
Sample mean of the predictor

The extra '1' inside the square root accounts for the natural scatter of individual observations around the true line, making the prediction interval always wider than the confidence interval for the conditional mean.