Correlation Notes

Measuring Linear Relationships With Correlation

1. Two variables form bivariate data when each individual provides a pair of values, written as $(x_1,y_1),(x_2,y_2),\dots,(x_n,y_n)$.
2. A scatter diagram (scatter plot) is usually the first representation because it lets you see whether an increase in $x$ tends to be associated with an increase or decrease in $y$.

Pearson's Correlation Coefficient Standardizes Covariance

1. From the previous article you should know that covariance which measures whether $x$ and $y$ tend to deviate from their means in the same direction, and its common "population-style" formula is $$s_{xy}=\frac{\sum (x-\bar{x})(y-\bar{y})}{n}$$
2. To make the strength of the relationship meaningful, we divide by the spread (standard deviation) of each variable.

Because we divide by $\sigma_x\sigma_y$, the coefficient $r$ has two key properties:

1. It is unit-free.
2. It always lies in the interval $-1\le r\le 1$.

How To Compute r From a Table of Values

For a set of $n$ pairs $(x,y)$, a practical workflow is:

1. Compute $\bar{x}$ and $\bar{y}$.
2. Compute $\sigma_x$ and $\sigma_y$ using $$\sigma_x=\sqrt{\frac{\sum (x-\bar{x})^2}{n}},\qquad \sigma_y=\sqrt{\frac{\sum (y-\bar{y})^2}{n}}$$
3. Compute $\frac{1}{n}\sum xy$.
4. Substitute into $$r=\frac{\frac{1}{n}\sum xy-\bar{x}\bar{{y}}}{\sigma_x\sigma_y}$$

On most calculators, $r$ is provided in the linear regression statistics menu.

Example

Calculating $r$ and Linking It to a Scatter Plot

Consider the bivariate data:

$x$	12	14	15	17	19
$y$	19	20	22	23	25

First calculate key statistics (rounded where needed):

• $n=5$
• $\sum x=77$, so $\bar{x}=77/5=15.4$
• $\sum y=109$, so $\bar{y}=109/5=21.8$
• $\sum xy=1709$, so $\frac{1}{n}\sum xy=341.8$

Covariance: $$s_{xy}=\frac{1}{n}\sum xy-\bar{x}\bar{y}=341.8-(15.4)(21.8)=6.08$$

Standard deviations: $$\sigma_x\approx 2.417,\qquad \sigma_y\approx 2.136$$

So the correlation coefficient is $$r=\frac{6.08}{(2.417)(2.136)}\approx 0.965$$

The scatter plot shows points clustered close to an upward-sloping line, consistent with a strong positive linear correlation.

Correlation Helps Describe Trends But Does Not Prove Causes

1. A strong correlation can justify drawing a line of best fit (by eye or using technology) and using it for interpolation, meaning prediction within the range of your data.
2. But there are important limitations.
3. Most importantly, correlation is not causation.
4. Two variables can be correlated because:
   1. one causes the other,
   2. they both depend on a third variable,
   3. or the apparent relationship is coincidence in a small sample.
5. Also be careful with extrapolation (predicting outside the observed $x$ values), because the relationship may change beyond the data range.

Correlation In Human Contexts: Attitudes and Scores

1. Correlation is not only for physical measurements (mass, length, temperature).
2. It is also used with scores and survey data.
3. For example, students may have a social media news satisfaction score and a television news satisfaction score.
4. A scatter plot of the paired scores can show whether students who rate one medium highly also rate the other highly.
5. If the computed $r$ is close to 0 (or negative), you might conclude there is little (or an inverse) linear relationship between the two attitudes, which can lead to discussion about preferences and identity.