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Forecast Accuracy Statistical Power Calculator

The Forecast Accuracy Statistical Power Calculator estimates the probability of detecting a specified change in a mean forecast-error metric with a two-sided normal test. It is useful when planning an evaluation of a forecasting method and you want to know whether the available number of out-of-sample error observations is likely to reveal a practically important improvement or deterioration relative to a benchmark mean.

Enter the number of error observations, the standard deviation of the error metric, the minimum mean difference you want to detect, and the significance level. The calculator expresses the difference as a standardized noncentral shift and estimates two-sided power. This model assumes approximately independent observations with a stable standard deviation; time-series dependence, paired model comparisons, repeated horizons, or changing variance may require a different design.

Forecast power inputs

Result
—
Approximate two-sided statistical power
Standard error—
Standardized shift—
Critical z—
Approx. beta—

1. Enter evaluation observations
Provide the number of comparable out-of-sample forecast-error observations available for the test.

2. Estimate error variability
Enter the standard deviation of the forecast-error metric under the evaluation design.

3. Define the minimum difference
Specify the absolute change in the mean error metric that would be practically important to detect.

4. Select significance
Choose the two-sided significance level. Lower alpha increases the evidence threshold and normally reduces power.

5. Review power and beta
The main result is estimated power. Beta in the detail panel is the approximate probability of failing to reject the null when the entered difference is truly present.

SE = σ / sqrt(n) Noncentral shift μ = Δ / SE = Δ × sqrt(n) / σ Critical value = z(1 − α/2) Power ≈ Φ(−z − μ) + 1 − Φ(z − μ)

n is the number of forecast-error observations, σ is their standard deviation, Δ is the absolute mean difference to detect, α is the two-sided significance level, and Φ is the standard normal cumulative distribution function.

What the result means

Power is the estimated probability that the test will flag a statistically significant mean change when the true change equals the entered detectable difference.

For paired forecasts from competing models, analyzing paired error differences is often more efficient than treating observations as an unpaired one-mean problem. Dependence across time can also lower effective power.

Given: 120 forecast-error observations, SD = 10 units, mean difference to detect = 3 units, alpha = 5%.

Calculation: SE = 10 / sqrt(120) ≈ 0.9129. Standardized shift μ = 3 / 0.9129 ≈ 3.286. Critical z = 1.96. Power ≈ Φ(−1.96 − 3.286) + 1 − Φ(1.96 − 3.286) ≈ 0 + 0.908.

Result: Approximate statistical power ≈ 90.8%.

Under the stated assumptions, the evaluation has a high chance of detecting a true 3-unit change in the mean forecast-error metric.

Why does lower forecast-error variability increase power?

A smaller standard deviation reduces the standard error of the mean. That makes the same absolute difference larger relative to sampling noise.

Should the detectable difference be positive or negative?

Enter the magnitude as a positive number. For a symmetric two-sided normal test, the power is the same for an equally sized increase or decrease under this simplified model.

Can I use the raw number of time points as n?

Only if those error observations provide roughly independent information. Serial correlation, overlapping horizons, and seasonality can make the effective sample size smaller.

How is beta related to power?

Beta is the probability of a type II error under the specified alternative. In this calculation, power = 1 − beta.

Is this suitable for comparing two forecast models on the same dates?

It can give a rough sense of signal-to-noise, but a paired analysis of per-date error differences is usually more appropriate because the two models are evaluated on the same outcomes.