Can an interval be asymmetric?

Yes, especially for skewed predictive distributions or transformed outcomes.

Is 95% always the right level?

No. The choice should reflect the purpose and consequences of missing the target.

Does collecting more history eliminate prediction uncertainty?

No; uncertainty about an average may shrink while new outcome variation remains.

Are prediction and tolerance intervals identical?

No. A tolerance interval targets coverage of a specified portion of a population with a stated confidence requirement.

AI Glossary / Time series and signal processing / Intermediate

Prediction interval

Definition

A prediction interval is a range constructed to cover a future observation, or another specified future quantity, at a stated probability or long-run coverage level under defined assumptions. It expresses uncertainty around a forecast. It differs from a confidence interval for an unknown mean because a future outcome includes its own random variation.

What it means

A point forecast gives one summary, such as an expected workload. A prediction interval adds a range intended to reflect plausible outcomes. Its meaning depends on the forecast target, horizon, information available, and statistical method.
A nominal 95% interval is not a promise that the next outcome will fall inside it. A calibrated frequentist procedure covers its target in 95% of repeated applications under its assumptions. A model-based predictive distribution can assign probability to a future outcome conditional on the model and information, with the validity of those assumptions remaining essential.

How it works

Specify what is being predicted: one observation, an average, a total, or a full future path. Estimate the relevant predictive uncertainty, including new random variation and parameter uncertainty where the method accounts for it. Obtain interval endpoints from the appropriate distribution or calibrated procedure.
For a Gaussian predictive distribution with known mean and standard deviation, a central 95% interval uses approximately the mean plus or minus 1.96 standard deviations. Other settings may require t distributions, simulation, bootstrap methods, or distributional quantiles. Skewed distributions often produce asymmetric intervals.
Forecast errors can be dependent across horizons. Separate 95% intervals at each horizon do not automatically provide 95% simultaneous coverage of the entire path. Model selection, structural change, uncertain future predictors, and omitted sources of variation can also make reported intervals too narrow.

Three practical examples

A call center uses a demand interval to assess capacity shortfalls rather than staffing solely for the point forecast.
A maintenance team reports a range for a future measurement and distinguishes it from uncertainty about the equipment's average operating level.
A budgeting analyst simulates a total across several months using their joint uncertainty instead of adding independently interpreted endpoint bounds.

Think of it this way

Think of estimating the average travel time and predicting tomorrow's particular journey. Learning the average very precisely does not eliminate tomorrow's traffic variation. An interval for the next trip must include that extra variation, while an interval for the average addresses a different question.

Worked example

Consider an original normal model with 25 independent observations, sample mean 100, and known population standard deviation 10. A future observation is independent of the sample and follows the same distribution. The mean is unknown, but the standard deviation is assumed known specifically for this illustration.
The standard error of the sample mean is 10/sqrt(25) = 2. A 95% confidence interval for the population mean is approximately 100 plus or minus 1.959964 times 2, or 96.0801 to 103.9199.
The prediction error for one new observation has variance 10 squared + 10 squared/25 = 104. Its standard deviation is sqrt(104), approximately 10.1980. The corresponding 95% prediction interval is 100 plus or minus 19.9878, or approximately 80.0122 to 119.9878.
The additional 100 units of variance come from the new observation itself. Independence justifies adding the variances. The normal quantile and arithmetic were independently checked. If the standard deviation were estimated rather than known, the appropriate normal-sample formula would use a t quantile and sample standard deviation. Serially dependent time-series observations require a model that accounts for that dependence.

Illustrative case study

Fictional case: Rosewood Services presents a narrow confidence interval around average demand as if it covered next week's actual demand. Its operations manager finds the distinction important because staffing must handle individual weeks, not only the long-run mean.
The analysts replace the label and calculate a prediction interval using a suitable demand model. They later track coverage and interval width across comparable forecast horizons. This invented case illustrates a reporting correction, not evidence that any interval method guarantees adequate staffing or handles every unusual event.

Common misconceptions and mistakes

Do not describe an interval for a mean as an interval for a new observation. Do not interpret the stated coverage as a guarantee for each subgroup, horizon, or operating regime unless the method supports that claim.
Do not compare methods only by narrowness. An interval can be attractively narrow because it ignores important uncertainty. Conversely, an excessively wide interval may achieve high coverage while offering little decision value.
Do not apply a symmetric normal formula automatically to counts, bounded outcomes, heavily skewed variables, or poorly specified models.

Business use

Prediction intervals help businesses evaluate contingency needs, service risks, and capacity choices. The appropriate action still depends on the relative cost of shortage and excess; a central interval is not automatically an optimal operating policy.
Record the target, horizon, coverage level, assumptions, and included uncertainty sources. Evaluate both coverage and sharpness on realistic later forecasts, using suitable interval scores where helpful. Monitor performance when the process or predictor distribution changes, and explain known limitations rather than presenting model uncertainty as certainty.

Check your understanding

Why is the prediction interval wider in the example?

Answer: It includes variation from the future observation as well as estimation of the mean.

What assumption permits adding the two variances?

Answer: Independence of the new observation and the sample mean.

Does the interval describe all future observations simultaneously?

Answer: No; it targets one new observation.

Would the same formula automatically work for dependent weekly demand?

Answer: No; the dependence must be modeled appropriately.

Related terms

Confidence interval
Calibration
Forecast horizon
forecast interval
predictive distribution
coverage
sharpness
quantile forecast
tolerance interval

Learn next

compare intervals for a mean, one observation, and a future average under the same simple model. Then evaluate forecast coverage separately by horizon and examine whether wider intervals correspond to more uncertain conditions.

Sources

Sources: NIST, predicting a single response: https://itl.nist.gov/div898/handbook/pmd/section5/pmd512.htm . Hyndman and Athanasopoulos, Forecasting: Principles and Practice, prediction intervals: https://otexts.com/fpp3/toolbox.html . Hyndman glossary: https://robjhyndman.com/mwh3/FG4.pdf . Primary passages consulted. Example and case are original.

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