Prediction Interval Calculator

A prediction interval is a range that is likely to contain the value of a single new observation at a given X value, based on a fitted regression model. Unlike a confidence interval for the mean response, a prediction interval accounts for both the uncertainty in estimating the regression line and the natural variability of individual data points, making it always wider.

First, fit a simple linear regression to get the slope (b₁) and intercept (b₀), then compute the predicted value Ŷ₀ = b₀ + b₁·X₀. Calculate the mean squared error (MSE = SSE / (n−2)) and use a t-distribution with n−2 degrees of freedom. The prediction interval is Ŷ₀ ± t(α/2, n−2) · √(MSE · (1 + 1/n + (X₀ − X̄)² / SSxx)), where SSxx is the sum of squared deviations of X.

A confidence interval estimates the range for the mean response (average Y) at a given X, while a prediction interval estimates the range for an individual new observation at that X. Because individual observations vary more than means, prediction intervals are always wider than confidence intervals for the same confidence level.

Set the confidence level to 0.95, which corresponds to α = 0.05. The t-critical value is taken at α/2 = 0.025 with n−2 degrees of freedom. The interval is then Ŷ₀ ± t(0.025, n−2) · SE_pred, where SE_pred = √(MSE · (1 + 1/n + (X₀ − X̄)² / SSxx)). This tool computes all of that automatically once you enter your data.

In Excel you can use LINEST() to get regression coefficients and residuals, then manually compute MSE and the standard error of prediction. Apply the T.INV.2T(0.05, n-2) function for the critical t-value and build the interval formula. However, using a dedicated prediction interval calculator like this one is much faster and less error-prone.

R² (the coefficient of determination) measures how much of the variability in Y is explained by the linear relationship with X. An R² of 1 means a perfect fit, while 0 means the model explains none of the variation. Higher R² values generally indicate a better-fitting regression model, though context matters.

You need at least 3 data points (n ≥ 3) to fit a simple linear regression and estimate a prediction interval, since the model uses n−2 degrees of freedom for the t-distribution. In practice, more data points produce more reliable intervals. With very few observations, the prediction interval will be very wide due to high uncertainty.

The prediction interval widens as X₀ moves farther from the mean X̄. This is because the term (X₀ − X̄)² / SSxx in the standard error formula grows larger, adding more uncertainty. Predictions made close to X̄ are more precise, while extrapolation far outside the range of your data produces very wide and less reliable intervals.