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.