Test Standard Error Calculator

Work out test standard error instantly with clear inputs, formula shown and shareable results.

About Test Standard Error Calculator

The standard error of measurement is sd x sqrt(1 - reliability), so it shrinks as reliability rises. With a 15 point standard deviation and reliability 0.90 the SEM is 4.74, meaning an observed score of 112 has a 95% confidence band from about 103 to 121 - a nineteen point span, which is why single scores should never be treated as exact.

Formulas

Standard error of measurement

SEM = sd x sqrt(1 - reliability)

Confidence band

score plus or minus 1.96 x SEM for 95%

Worked example

With the default values below, Test Standard Error Calculator returns Standard Error Of Measurement of 4.743. Change any input above and every figure here is recalculated in your browser.

Inputs used

Standard deviation of test scoresminimum 0: 15
Test reliability coefficientaccepts 0–1: 0.9
An observed score: 112

Results

Standard Error Of Measurement: 4.743
Lower Bound Of The 68% Band: 107.26
Upper Bound Of The 95% Band: 121.30
Width Of The 95% Band: 18.59

Step by step

1Values used
2Standard error of measurement
3Confidence band
4Standard Error Of Measurement= 4.743
5Lower Bound Of The 68% Band= 107.26
6Upper Bound Of The 95% Band= 121.30
7Width Of The 95% Band= 18.59

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 3 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Why does a reliable test still have a wide band?

Because even reliability of 0.90 leaves 10% of variance as error. Halving the SEM requires reliability of 0.975, which is very hard to achieve.

Should the band be applied to the observed score?

Strictly it should be centred on the estimated true score, which regresses toward the mean. Centring on the observed score is the common simplification.