Thorndike's Calculator for Correlation Correction

Thorndike Case 2 - Adjustment for Range Restriction Using Standard Deviations

Thorndike’s Case 2 formula is a widely used method for adjusting correlation coefficients when a sample has undergone range restriction. Range restriction occurs when the variability in a sample is smaller than that of the population, leading to underestimation of the true correlation between two variables. In particular, when a researcher selects a subset of data (e.g., only the top performers on a test), the restricted sample does not capture the full variability of the population, which distorts the observed correlation.

The adjustment for range restriction in Thorndike’s Case 2 is done using the ratio of the standard deviations of the unrestricted population and the restricted sample. This correction allows researchers to estimate what the correlation would be in the unrestricted population, given the observed correlation in the restricted sample.

The corrected correlation is calculated as:

\[ r_{xy}^c = \frac{r \cdot \frac{s_u}{s_r}}{\sqrt{1 - r^2 + r^2 \cdot \left( \frac{s_u}{s_r} \right)^2 }} \]

Where:

• \( r \) is the observed correlation in the restricted sample
• \( s_u \) is the standard deviation of the unrestricted population
• \( s_r \) is the standard deviation of the restricted sample
• \( r_{xy}^c \) is the corrected correlation, which estimates the true correlation in the unrestricted population

This formula corrects the attenuation in correlation caused by range restriction. The ratio \( \frac{s_u}{s_r} \) accounts for the difference in variability between the unrestricted and restricted samples. The corrected correlation, \( r_{xy}^c \), reflects what the correlation would have been if the full range of data had been observed, without restriction.

Thorndike's Case 2 is particularly useful in applied research contexts such as personnel selection, educational testing, and any scenario where only a subset of the full population is available for analysis due to selection or other filtering processes.

By applying this correction, researchers can ensure that their analyses yield more accurate and representative estimates of relationships between variables, reducing bias introduced by restricted samples.