Skip to content

Business Mathematics and Statistics · Ch 9 — Correlation and Regression Analysis

Karl Pearson's Coefficient of Correlation

2

Karl Pearson's Coefficient of Correlation

Karl Pearson's coefficient of correlation, usually written rr, puts an exact, unit-free number on the strength and direction of a linear relationship between two variables.

Note

Karl Pearson's Correlation Coefficient

r=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2∑(yi−yˉ)2r = \dfrac{\sum(x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum(x_i-\bar x)^2}\sqrt{\sum(y_i-\bar y)^2}}

Equivalently, using raw sums directly (the 'direct method', often faster by hand):

r=n∑xiyi−(∑xi)(∑yi)n∑xi2−(∑xi)2n∑yi2−(∑yi)2r = \dfrac{n\sum x_iy_i - (\sum x_i)(\sum y_i)}{\sqrt{n\sum x_i^2-(\sum x_i)^2}\sqrt{n\sum y_i^2-(\sum y_i)^2}}

Both forms always give the identical value for the same data; the deviation form makes clear why the formula works (the numerator is positive when xx and yy tend to be above their own means together, negative when one tends to be above while the other is below), while the direct method avoids computing deviations at all, which is usually quicker for hand calculation.

Note

Properties of r

  • rr always lies between −1-1 and +1+1 inclusive: −1≤r≤1-1\le r\le1.
  • r=+1r=+1: perfect positive correlation (every point lies exactly on an upward-sloping line).
  • r=−1r=-1: perfect negative correlation (every point lies exactly on a downward-sloping line).
  • r=0r=0: no linear correlation (though a strong non-linear relationship could still exist — rr specifically measures only the linear part of the relationship). …
Definition 3Karl Pearson's Coefficient of Correlation (r)

A unit-free measure of the strength and direction of the linear relationship between two variables, always between −1-1 and +1+1; $r=\sum(x-\bar x)(y-\bar y)/\sqrt{\sum( …

Start your 14-day free trial to unlock the full solution →
← Previous · 1Bivariate Data and the Scatter DiagramNext · 3 →Spearman's Rank Correlation Coefficient