Why Pearson Correlation Is Unaffected by Different Measurement Units

Summary

The question asks whether a return measured as a percentage can be correlated with a variable measured in dollars or basis points, and whether standardizing is necessary. The answers explain that Pearson correlation already divides covariance by each variable’s standard deviation, making the result invariant to changes of scale. Computing it from standardized variables is therefore equivalent to using the original measurements, provided the transformations are linear rescalings.

One answer cautions that standardization can matter when comparing variables whose scales or variances differ substantially in other contexts, while the other emphasizes that it does not change Pearson correlation itself. The discussion is brief and gives no empirical example. Its conclusion applies specifically to Pearson correlation; it does not address other dependence measures, nonlinear transformations, data quality, or whether correlation is an appropriate analysis for a particular trading question.

Key ideas

• Pearson correlation is unchanged when either variable is rescaled by a positive constant.
• Its definition is equivalent to taking the expectation of the product of standardized variables.
• Different units, such as percentage returns and dollar changes, do not by themselves require standardization before calculating Pearson correlation.
• The discussion does not establish that correlation is appropriate or informative for every pair of variables.

Tags

# Correlation with Differ Units of Measurement


# Correlation with Differ Units of Measurement












I was wondering how to accurately get the correlation between a variable of percent change return and a variable of dollar change or basis points. Should I standardize both variables or will that lose relevant information? I would appreciate any guidance.

## Answer by Quant In Spe (score 1)

https://quant.stackexchange.com/a/71402

When calculating a correlation it is generally advised to standardize, yes. I don't see why this case would be an exception although failing to standardize wouldn't probably be the biggest issue here. This is because the variance of the two variables are fairly comparable.

It is very important to standardize when you're looking to get a correlation between some percent change and the level yearly revenue of a company in dollars, for example.

## Answer by Kermittfrog (score 1)

https://quant.stackexchange.com/a/71412

When calculationg the Pearson correlation coefficient, it does not matter. It is defined as

$$ \rho\equiv \frac{cov(x,y)}{\sigma_x\sigma_y}=\frac{\mathrm{E}\left((x-\mu_x)(y-\mu_y)\right)}{\sigma_x\sigma_y}=\mathrm{E}\left(\frac{x-\mu_x}{\sigma_x}\frac{y-\mu_y}{\sigma_y}\right)=\mathrm{E}\left(z_xz_y\right) $$

where $z_x,z_y$ are standardised.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.