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Standard Error of Measurement Calculator and Test Score Banding

How do you calculate the standard error of measurement?

SEM = SD x √(1 - rxx), where SD is the standard deviation of the test scores and rxx its reliability. With SD 10 and reliability .85, SEM = 3.87. Two candidates' scores differ reliably only if the gap exceeds C x SEM x √2: 10.7 points at 95%, which is the basis of score banding in selection.

Ranking candidates by test score treats a two-point gap as real. With an ordinary reliability of .85 and an SD of 10, scores up to 10.7 points apart cannot be told apart at 95% confidence.

Enter the SD, reliability and a list of candidates; the calculator gives the SEM, the band, who falls in it, and each person's estimated true score.

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Built by Muhammad Shahzeb Anwar, MSc Industrial and Organizational Psychology. Banding comes from personnel selection research (Cascio, Outtz, Zedeck and Goldstein, 1991); it is where psychometrics meets hiring decisions.

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Alpha or test-retest, between 0 and 1.
Example scores are filled in; replace them.

SEM, SED and the band

The SEM describes the error around one score. Comparing two people involves two errors, so the standard error of the difference is SEM x √2. A band of width C x SED below the top score (C = 1.96 for 95%) contains every score that is not reliably different from it. In a fixed band everyone in it is treated as equal; in a sliding band, once the top scorer is selected the band is recalculated from the next highest score.

Why the true-score interval is centred differently

A confidence interval for a person's true score is not observed score +/- C x SEM. Dudek (1979) and Harvill (1991) show it should centre on the estimated true score, mean + rxx x (score - mean), which is pulled toward the mean, with a standard error of SD x √(rxx(1 - rxx)). The calculator shows both, because textbooks often give only the first.

Before you rely on this

What this will not tell you

Banding decides who is indistinguishable on the test; how to choose within the band is a separate policy decision. In the United States, adjusting scores or using different cut-offs by race, colour, religion, sex or national origin is prohibited (Civil Rights Act 1991, s.106); choosing within a band on other job-related criteria is a different question that needs legal advice. Nothing you type leaves your browser.

Questions

Questions people actually ask

What is the formula for the standard error of measurement?

SEM = SD x the square root of (1 - reliability). With an SD of 10 and reliability .85, SEM is 3.87.

What is the standard error of the difference?

The standard error for the difference between two people's scores: SEM x the square root of 2. Two scores differ reliably at 95% only if they are more than 1.96 x SED apart.

What is banding in personnel selection?

Treating candidates whose scores fall within a statistically defined distance of the top score as equivalent, because the test cannot reliably separate them.

What is the difference between fixed and sliding bands?

A fixed band is set once from the top score. A sliding band moves down each time the top scorer is selected, recalculated from the highest remaining score.

How does reliability affect the SEM?

Lower reliability means a larger SEM and wider bands. At reliability .70 and SD 10, SEM is 5.48 and the 95% band is over 15 points wide.

The papers behind the method

Sources

Each reference was checked against its DOI record on 17 October 2026.

  1. Cascio, W. F., Outtz, J., Zedeck, S., & Goldstein, I. L. (1991). Statistical implications of six methods of test score use in personnel selection. Human Performance, 4(4), 233-264: https://doi.org/10.1207/s15327043hup0404_1
  2. Dudek, F. J. (1979). The continuing misinterpretation of the standard error of measurement. Psychological Bulletin, 86(2), 335-337: https://doi.org/10.1037/0033-2909.86.2.335
  3. Harvill, L. M. (1991). Standard error of measurement. Educational Measurement: Issues and Practice, 10(2), 33-41: https://doi.org/10.1111/j.1745-3992.1991.tb00195.x
Citation

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Free to use and free to quote. If a figure here is useful in something you are writing, a link back is all that is asked.

Muhammad Shahzeb Anwar, MSc Industrial and Organizational Psychology. Standard Error of Measurement and Banding Calculator. Shahzeb Speaks. https://www.shahzebspeaks.com/standard-error-of-measurement-banding-calculator/

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Interpreting one person's score?

The paid Test Score Interpreter turns a single score into a percentile, z, T and stanine, with the band around it and how far apart two scores must be to differ. It runs offline in one file. This free page does the selection banding of a candidate list.

See the Test Score Interpreter