Why are the diagonal values in a correlation matrix often replaced with dashes when simplifying the table for readability?

Correlation matrices are often simplified for better readability by filling only half of the table, as the other half would contain identical values. Furthermore, since any variable's correlation with itself is always $$1.00$$, these redundant diagonal values are often replaced with dashes.

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A correlation matrix is a specific type of table utilized to present the correlation coefficients—typically Pearson’s $$r$$—among several different variables. To avoid redundant information and improve overall readability, only the lower half of the matrix is filled with values, as the upper half would display identical mirror correlations. Furthermore, because any variable's correlation with itself is always a perfect $$+1.00$$, these values along the diagonal are conventionally replaced with dashes.

Correlation Matrix

https://kpu.pressbooks.pub/psychmethods4e/

KPU Research Methods in Psychology - 4th American Edition

A correlation matrix concisely presents Pearson's $$r$$ values among multiple variables, such as working memory, executive function, and age. The variables are listed sequentially down the leftmost column and across the top headings. Because any variable perfectly correlates with itself ($$r = 1.00$$), the diagonal cells are conventionally filled with dashes. To prevent redundant information, the table only displays the computed correlation coefficients in the bottom half of the matrix, leaving the mirrored upper half blank.

Example of an APA-Style Correlation Matrix

To evaluate the validity of their new Need for Cognition Scale, Cacioppo and Petty utilized a complex correlational approach. They measured participants' scores on the new scale alongside three other established variables—intelligence, socially desirable responding, and dogmatism—and constructed a correlation matrix; the resulting pattern of correlations supported their theoretical predictions and thus the validity of the new measure.

Example of Assessing Validity via Correlation Matrix

Simplifying Correlation Matrices

A correlation matrix can present the relationships among several variables. For example, a study by McCabe and colleagues used a correlation matrix to show relationships between working memory and other cognitive variables. The matrix revealed an extremely strong correlation ($$.96$$) between working memory and executive function, a medium correlation ($$.27$$) between working memory and vocabulary, and showed that all measures except vocabulary tend to decline with age.

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