Spearman–Brown Prophecy Calculator (Test Length & Reliability)

How it works

r_new = n × r ÷ [1 + (n − 1) × r], n = new length ÷ old length; n needed = r_target(1 − r) ÷ [r(1 − r_target)]; split-half correction uses n = 2

Adding parallel items to a test averages out more of the random error in each, so reliability rises in a predictable way described by the Spearman–Brown formula: doubling a 0.70 test gives 0.82, quadrupling 0.90, with each step yielding less as the ceiling of 1.0 approaches. The same algebra corrects a split-half correlation — which compares two half-length tests — up to the full-test value, and it tells a researcher how many raters must be averaged to make ratings trustworthy or how many measurements to take of a noisy quantity. The prophecy holds only for additions of the same quality; the fastest route to reliability is usually replacing non-discriminating items rather than piling on more.

Worked example

A 30-item test with reliability 0.72 lengthened to 50 items (factor 1.67) should reach 0.81; hitting 0.85 would need about 67 items (2.2×). As a split-half correction, a half-test correlation of 0.72 implies a full-test reliability of 0.84; averaging three raters who agree at 0.72 gives a rating reliability of 0.89.

Frequently asked questions