## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 6, Number 3 (1996), 992-1005.

### Rank inversions in scoring multipart examinations

#### Abstract

Let $(X_i, Y_i), i = 1, \dots, n$, be independent random vectors
with a standard bivariate normal distribution and let $s_X$ and $s_Y$ be the
sample standard deviations. For arbitrary $p, 0 < p < 1$, define $T_i =
pX_i + (1 - p) Y_i$ and $Z_i = pX_i / s_X + (1 - p) Y_i / s_Y, i = 1, \dots,
n$. The couple of pairs $(T_i, Z_i)$ and $(T_j, Z_j)$ is said to be discordant
if either $T_i < T_j$ and $Z_i > Z_j$ or $T_i > T_j$ and $Z_i <
Z_j$. It is shown that the expected number of discordant couples of pairs is
asymptotically equal to $n^{3/2}$ times an explicit constant depending on
*p* and the correlation coefficient of $X_i$ and $Y_i$. By an application
of the Durbin-Stuart inequality, this implies an asymptotic lower bound on
the expected value of the sum of $(\rank(Z_i) - \rank(T_i))^+$. The problem
arose in a court challenge to a standard procedure for the scoring of multipart
written civil service examinations. Here the sum of the positive rank
differences represents a measure of the unfairness of the method of
scoring.

#### Article information

**Source**

Ann. Appl. Probab., Volume 6, Number 3 (1996), 992-1005.

**Dates**

First available in Project Euclid: 18 October 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1034968237

**Digital Object Identifier**

doi:10.1214/aoap/1034968237

**Mathematical Reviews number (MathSciNet)**

MR1410125

**Zentralblatt MATH identifier**

0866.62031

**Subjects**

Primary: 62F07: Ranking and selection 62H20: Measures of association (correlation, canonical correlation, etc.)

**Keywords**

Bivariate normal distribution concordance ranks sample variance

#### Citation

Berman, Simeon M. Rank inversions in scoring multipart examinations. Ann. Appl. Probab. 6 (1996), no. 3, 992--1005. doi:10.1214/aoap/1034968237. https://projecteuclid.org/euclid.aoap/1034968237