Range Restriction Calculator
Why does my selection test show such a low validity?
Because you only measured the people you hired. Everyone who scored badly was screened out, so the spread of scores in your data is narrower than in the applicant pool, and a narrower spread mechanically shrinks a correlation. On the figures here, an observed validity of 0.25 among hires corresponds to about 0.40 in the applicant pool — the test is being 58% understated by the very selection it was used for.
You ran the numbers on your own hires and the test barely correlates with performance. That is what a working test looks like once selection has already happened.
The people who would have proved the test right were never hired, so their performance was never measured.
Built by Muhammad Shahzeb Anwar, MSc Industrial and Organizational Psychology, who also builds the paid tools in this shop. Every in-house validity study has this problem built into it, and correcting for it is standard practice in the research it gets compared against.
What the validity would be in the full applicant pool
You need the spread of predictor scores in two groups: everyone who applied, and the ones you hired.
Why selection makes a good test look bad
To measure whether a test predicts performance you need both numbers for the same people: their test score, and how they did in the job. You can only have that for people you hired.
But you did not hire at random. You hired the higher scorers. So the scores in your data cover a narrow band near the top, and the low scorers — the ones whose poor performance would have demonstrated that the test works — are missing, because they were never given the chance to perform badly.
A correlation measures how much two things move together across a range. Cut the range down and the correlation falls, even though the underlying relationship has not changed at all. That is range restriction, and every in-house validity study has it.
What the correction does
Thorndike's Case II formula rescales the correlation back to what it would have been in the full applicant pool. It needs one extra fact: how much wider the spread of scores was among applicants than among hires.
rcorrected = r × U / √(1 − r² + r²U²), where U is the applicant spread divided by the hire spread. At U = 1 nothing changes. The bigger the difference between the two spreads, the larger the correction.
This is not a trick to make results look better. Published validity figures — the meta-analyses your test vendor quotes — are routinely corrected this way, so comparing your uncorrected figure against their corrected one is not a fair comparison in the first place.
What it cannot do
It cannot create validity. Put an observed correlation of zero in and you get zero out, however restricted the range was. The correction rescales a relationship; it does not conjure one.
It assumes the restriction happened directly on this predictor, and that the relationship is linear and equally spread across the range. If people were selected mainly on an interview and the test was incidental, this is the wrong correction — that is indirect restriction, and it needs a different formula.
And a corrected figure is an estimate, not a measurement. Report it as corrected, say which formula, and give the observed figure beside it. A validity quoted without saying whether it has been corrected is not interpretable.
What this will not tell you
This is the correction for direct restriction on the predictor itself. If selection was made on something else that merely correlates with your test, the restriction is indirect and this formula understates it. It also cannot fix a study that is too small: a correlation from twenty hires is unstable before any correction is applied, and correcting it only makes an unstable number larger.
Questions people actually ask
What is range restriction in selection testing?
It is the shrinking of a correlation that happens because you can only measure job performance for the people you hired, and you hired the higher scorers. The narrow spread of scores in that group mechanically reduces the correlation.
What is the Thorndike Case II correction?
A formula that rescales an observed validity back to the full applicant pool: r x U / sqrt(1 - r^2 + r^2 x U^2), where U is the applicant score spread divided by the hire score spread.
Can correcting for range restriction make a bad test look good?
No. An observed correlation of zero corrects to zero however restricted the range. The correction rescales an existing relationship rather than creating one.
Should I report the corrected or the uncorrected validity?
Both, clearly labelled. Published meta-analytic validities are usually corrected, so comparing your uncorrected figure against a vendor's corrected one is not a like-for-like comparison.
Cite this tool
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. Range Restriction Calculator. Shahzeb Speaks. https://www.shahzebspeaks.com/range-restriction-calculator/
The other free tools
Each one answers a single question and refuses the ones it cannot. All free, none asks for an email address.
Two candidates, six points apart. Same score?
The paid Test Score Interpreter takes the other half of this problem: what a single score means once measurement error is admitted. On a test with a reliability of .85 two candidates have to be 16.1 points apart before the difference is real, which is further apart than most shortlists assume.
See the paid tool