Prediction Intervals

A confidence interval answers "where does the true average likely lie?" A prediction interval answers a different, harder question: "where will the next single observation likely fall?" Because any one observation carries both the uncertainty about the average and its own individual randomness around that average, a prediction interval is always at least as wide as, and usually noticeably wider than, the corresponding confidence interval.

For a simple linear regression forecast, the prediction interval's width comes from adding the variance of the estimated mean to the residual variance of individual points, then taking the square root, the residual variance is the piece a confidence interval on the mean ignores entirely.

Worked example. A model forecasts next quarter's revenue growth with a point estimate of 8%. The 95% confidence interval on the average growth rate across many similar quarters is 8% ± 1%, i.e. 7% to 9%. But the 95% prediction interval for this specific upcoming quarter is 8% ± 6%, i.e. 2% to 14%, because a single quarter's actual growth bounces around far more than the long-run average does.

Reporting a confidence interval when a prediction interval is what's needed is a common and consequential mistake in forecasting, it makes a single forecast look far more precise than the data actually supports, which matters directly when the forecast is being used to size a position or set a risk limit.