Prediction Interval Calculator

Estimate a person's likely score on a second, correlated test (e.g. a retest, or a different IQ test) from their score on a first test. Enter the mean, SD, and correlation between the two tests, then choose either a percent of people you want the interval to capture, or a fixed interval width, and the other is computed automatically.

Prediction Interval Calculator

Enter test parameters, then a percent captured OR an interval width.

Mean

Correlation (r)

Test 1 score

Solve for percent captured or width

% captured

± width

About prediction intervals

When two tests are correlated but not identical, a score on Test 1 does not tell you someone's exact score on Test 2 — it tells you a predicted score and a range around it. Because of regression to the mean, that predicted score is pulled toward the mean rather than equal to X. This calculator uses the mean, SD, and correlation (r) between the two tests to compute that predicted score and the interval around it, based on the normal distribution of residuals from linear regression.

Predicted score \[ \hat{Y} = \mu + r\,(X - \mu) \] Residual standard deviation \[ \sigma_{\text{resid}} = \sigma\sqrt{1 - r^{2}} \] Prediction interval \[ \hat{Y} \pm z\,\sigma_{\text{resid}} \] Notation \(\mu \) and \(\sigma \) are the mean and standard deviation of the test scale, \(X \) is the Test 1 score, and \(r \) is the correlation between the two tests. \(z \) is the standard-normal critical value for the chosen % captured — e.g. \(z \approx 1.96\) for 95%, \(z \approx 1.282\) for 80%, \(z \approx 1.00\) for 68.3%.

To use this tool, enter either a % captured or a ± width, but not both. The Test 1 score is optional — leave it blank to center the interval on the mean.