Tied ranks correction
Also called: Repeated ranks · Topic: Economic Data: Census, NSS, Surveys and Statistical Tools · NCERT: Class 11, Ch 6 "Correlation"
Meaning
In Spearman's rank correlation, two or more items sometimes have the same value. These are called tied ranks, or repeated ranks. Each tied item gets the mean of the ranks those items would otherwise have taken. Ties distort the result, so a correction is made. For each group of ties, (m³ − m)/12 is added to ΣD², where m is the number of items tied. The formula becomes: rs = 1 − 6[ΣD² + Σ(m³ − m)/12] / (n³ − n), where D is the difference in ranks and n is the number of pairs.
Example
Two students tie for the 3rd and 4th places, so each gets rank 3.5. With m = 2, the correction is (8 − 2)/12 = 0.5. If three students tie for the 5th, 6th and 7th places, each gets rank 6. With m = 3, the correction is (27 − 3)/12 = 2. Both corrections are added to ΣD².
Don't confuse with
- Plain Spearman formula, rs = 1 − 6ΣD²/(n³ − n): this is correct only when there are no ties.