Work out test standard error instantly with clear inputs, formula shown and shareable results.
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.
Standard error of measurement
SEM = sd x sqrt(1 - reliability)
Confidence band
score plus or minus 1.96 x SEM for 95%
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.
Strictly it should be centred on the estimated true score, which regresses toward the mean. Centring on the observed score is the common simplification.