A Pearson correlation is a number between -1 and 1 that indicates the extent to which two variables are linearly related.
The Pearson correlation is also known as the “product moment correlation coefficient” (**PMCC**) or simply “**correlation**”.

Pearson correlations are suitable only for metric variables (which include dichotomous variables).

- For
**ordinal variables**, use the Spearman correlation or Kendall’s tau and - for
**nominal variables**, use Cramér’s V.

## Correlation Coefficient - Example

We asked 40 freelancers for their yearly incomes over 2010 through 2014. Part of the raw data are shown below.

Today’s question is:
is there any relation between income over 2010

and income over 2011?
Well, a splendid way for finding out is inspecting a scatterplot for these two variables: we'll represent each freelancer by a dot. The horizontal and vertical positions of each dot indicate a freelancer’s income over 2010 and 2011. The result is shown below.

Our **scatterplot shows a strong relation** between income over 2010 and 2011: freelancers who had a low income over 2010 (leftmost dots) typically had a low income over 2011 as well (lower dots) and vice versa. Furthermore, **this relation is roughly linear**; the main pattern in the dots is a straight line.

The extent to which our dots lie on a straight line indicates the strength of the relation. The Pearson correlation is a number that indicates the exact strength of this relation.

## Correlation Coefficients and Scatterplots

A correlation coefficient indicates the extent to which dots in a scatterplot lie on a straight line. This implies that we can usually estimate correlations pretty accurately from nothing more than scatterplots. The figure below, in which the **correlation coefficient is denoted by “r”** nicely illustrates this point.

## Correlation Coefficient - Basics

Some basic points regarding correlation coefficients are nicely illustrated by the previous figure. The least you should know is that

**Correlations are never lower than -1.**A correlation of -1 indicates that the data points in a scatter plot lie exactly on a straight descending line; the two variables are perfectly negatively linearly related.- A
**correlation of 0**means that two variables don't have any linear relation whatsoever. However, some non linear relation may exist between the two variables. **Correlation coefficients are never higher than 1.**A correlation coefficient of 1 means that two variables are*perfectly*positively linearly related; the dots in a scatter plot lie exactly on a straight ascending line.

## Correlation Coefficient - Interpretation Caveats

When interpreting correlations, you should keep some things in mind. An elaborate discussion deserves a separate tutorial but we'll briefly mention two main points.

- Correlations may or may not indicate
**causal relations**. Reversely, causal relations from some variable to another variable may or may not result in a correlation between the two variables. - Correlations are very sensitive to
**outliers**; a single unusual observation may have a huge impact on a correlation. Such outliers are easily detected by a quick inspection a scatterplot.

## Correlation Coefficient - Software

Most spreadsheet editors such as Excel, Google sheets and OpenOffice can compute correlations for you. The illustration below shows an example in Googlesheets.

## Correlation Coefficient - Correlation Matrix

Keep in mind that correlations apply to pairs of variables. If you're interested in more than 2 variables, you'll probably want to take a look at the correlations between all different variable pairs. These correlations are usually shown in a square table known as a **correlation matrix**. Statistical software packages such as SPSS create correlations matrices before you can blink your eyes. An example is shown below.

Note that the **diagonal elements** (in red) are the correlations between each variable and itself. This is why they are always 1.

Also note that the **correlations beneath the diagonal** (in grey) are redundant because they're identical to the correlations above the diagonal. Technically, we say that this is a symmetrical matrix.

Finally, note that the pattern of correlations makes perfect sense: correlations between yearly incomes become lower insofar as these years lie further apart.

## Pearson Correlation - Formula

If we want to inspect correlations, we'll have a computer calculate them for us. You'll rarely (probably never) need the actual formula. However, for the sake of completeness, a Pearson correlation between variables X and Y is calculated by

$$r_{XY} = \frac{\sum_{i=1}^n(X_i - \overline{X})(Y_i - \overline{Y})}{\sqrt{\sum_{i=1}^n(X_i - \overline{X})^2}\sqrt{\sum_{i=1}^n(Y_i - \overline{Y})^2}}$$

The formula basically comes down to dividing the covariance by the product of the standard deviations. Since a coefficient is a number divided by some other number our formula shows why we speak of a correlation *coefficient*.

## Correlation - Statistical Significance

The data we've available are often -but not always- a small sample from a much larger population. If so,
we may find a non zero correlation in our sample

even if it's zero in the population. The figure below illustrates how this could happen.

If we ignore the colors for a second, all 1,000 dots in this scatterplot visualize some population. The **population correlation -denoted by ρ- is zero** between test 1 and test 2.

Now, we *could* draw a sample of N = 20 from this population for which the correlation **r = 0.95**.
Reversely, this means that a sample correlation of 0.95 doesn't prove with certainty that there's a non zero correlation in the entire population. However, finding r = 0.95 with N = 20 is **extremely unlikely** if ρ = 0. But precisely how unlikely? And how do we know?

## Correlation - Test Statistic

If ρ -a population correlation- is zero, then the probability for a given sample correlation -its statistical significance- depends on the sample size. We therefore combine the sample size and r into a single number, our test statistic t:
$$T = R\sqrt{\frac{(n - 2)}{(1 - R^2)}}$$

Now, T itself is not interesting. However, we need it for finding the significance level for some correlation. T follows a t distribution with ν = n - 2 degrees of freedom but only if some assumptions are met.

## Correlation Test - Assumptions

The statistical significance test for a Pearson correlation requires 3 assumptions:

**independent observations**;- the population correlation,
**ρ = 0**; **normality**: the 2 variables involved are bivariately normally distributed in the population. However, this is not needed for a reasonable sample size -say, N ≥ 20 or so.The reason for this lies in the central limit theorem.

## Pearson Correlation - Sampling Distribution

In our example, the sample size N was 20. So if we meet our assumptions, T follows a t-distribution with df = 18 as shown below.

This distribution tells us that there's a 95% probability that -2.1 < t < 2.1, corresponding to -0.44 < r < 0.44. Conclusion:
if N = 20, there's a 95% probability of finding -0.44 < r < 0.44.
There's only a 5% probability of finding a correlation outside this range. That is, such correlations are statistically significant at α = 0.05 or lower: they are (highly) unlikely and thus refute the null hypothesis of a zero population correlation.

Last, our sample correlation of 0.95 has a p-value of 1.55e^{-10} -one to 6,467,334,654. We can safely conclude there's a non zero correlation in our entire population.

Thanks for reading!

## This tutorial has 16 comments

## By Ruben Geert van den Berg on March 30th, 2019

Hi Kaan!

We usually don't know the population correlation. So we start off with a

guessthat it's zero -the null hypothesis. Given this guess, we can compute the probability for a given sample correlation -the significance level. And if this probability is very small, we reject our initial guess of a zero population correlation.This reasoning may need to sink in for a few days. But does it make any more sense now?