Prediction Interval Calculator

A prediction interval tells you where a single new observation is likely to fall, given a fitted regression line. It is wider than a confidence interval for the mean response because it has to account for two sources of uncertainty: the uncertainty in where the true mean response is, and the random scatter of individual observations around that mean.

The formula for a prediction interval at a new x value x* is:

y-hat +/- t * s * sqrt(1 + 1/n + (x* - x-bar)^2 / Sxx)

where y-hat is the predicted value from the regression line, s is the residual standard error, n is the sample size, x-bar is the mean of the x values in the training data, Sxx is the sum of squared deviations of x, and t is the t-critical value with n-2 degrees of freedom.

The extra 1 inside the square root is what makes prediction intervals wider than confidence intervals for the mean. The confidence interval for the mean response uses sqrt(1/n + (x* - x-bar)^2 / Sxx), which captures uncertainty in the mean regression line but not observation-level scatter.

The term (x* - x-bar)^2 / Sxx grows as x* moves away from the center of the data. Predictions near x-bar are more reliable than predictions at the extremes. This is why extrapolating a regression line far outside the data range produces very wide intervals and is generally inadvisable.

To use this calculator: fit a linear regression in your statistics software, record the residual standard error (often labeled s or RMSE), the sample size, and the Sxx from the summary. Then plug in any new x* to get the interval.

A 95% prediction interval means: if you repeated the study many times, about 95% of the intervals you compute would contain the actual new observation. It does not mean there is a 95% chance the new observation falls in this specific interval.