# Thorndike's Pearson Correlation Correction for Range Restriction

## Thorndike Case 2 - Correction for Range Restriction

Thorndike's Case 2 formula (1947) is a pivotal tool for correcting correlation coefficients for range restriction, an essential process for researchers and statisticians aiming for accurate data interpretation. This correction compensates for the reduced variability seen in samples that do not capture the full spectrum of data present in the population, thus refining the accuracy of correlation assessments.

Range restriction occurs when the data collected for analysis do not reflect the entire scope or variability of the population. This limitation can significantly distort the correlation coefficient, leading to inaccurate conclusions about the relationship between variables.

Correcting correlation coefficients for range restriction is crucial in research and statistical analysis. It ensures that the calculated correlation more accurately reflects the true relationship between variables across the full range of data.

The formula for Thorndike Case 2 is expressed as:

\[ r_{xy}^c = \frac{r_{xy}}{\sqrt{(1 - b^2_y)(1 - b^2_x)}} \]Where:

- \( r_{xy} \) is the observed correlation coefficient between variables \( X \) and \( Y \)
- \( b_y \) and \( b_x \) represent the selection ratios for variables \( Y \) and \( X \), respectively
- \( r_{xy}^c \) is the corrected correlation coefficient, offering a more accurate representation of the variables' relationship in the absence of range restriction

This correction not only enhances the validity of research findings but also supports robust data-driven decision-making.

## References

Thorndike, R. L. (1947). *Research problems and techniques* (Rep. No. 3 AAF Aviation Psychology Program Research Reports). Washington, DC: U.S. Government Printing Office.