Statistics Definitions > Correlation Coefficient Formula

Also see: Correlation Coefficient (How to Find it).

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## Correlation Coefficient Formula: Overview.

Correlation coefficient formulas are used to find how strong a relationship is between data. The formulas return a value between -1 and 1, where:

- 1 indicates a strong positive relationship.
- -1 indicates a strong negative relationship.
- A result of zero indicates no relationship at all.

## Meaning.

- A correlation coefficient of 1 means that for every positive increase of 1 in one variable, there is a positive increase of 1 in the other.
- A correlation coefficient of -1 means that for every positive increase of 1 in one variable, there is a negative decrease of 1 in the other.
- Zero means that for every increase, there isn’t a positive or negative increase. The two just aren’t related.

The absolute value of the correlation coefficient gives us the relationship strength. The larger the number, the stronger the relationship. For example, |-.75| = .75, which has a stronger relationship than .65.

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## Types of correlation coefficient formulas.

There are several types of correlation coefficient formulas.

One of the most commonly used formulas in stats is Pearson’s correlation coefficient formula. In fact, if you’re taking a basic stats class, this is the one you’ll probably use:

Two other formulas are commonly used: the sample correlation coefficient and the population correlation coefficient.

## Sample correlation coefficient

S_{x} and s_{y} are the sample standard deviations, and s_{xy} is the sample covariance.

## Population correlation coefficient

The population correlation coefficient uses σ_{x} and σ_{y} as the population standard deviations, and σ_{xy} as the population covariance.

Other similar formulas you might come across that involve correlation:

- Intraclass Correlation.
- Kendall’s Tau.
- Moran’s I.
- Partial Correlation.
- Point Biserial Correlation.
- Polychoric Correlation.
- Spearman Rank Correlation.
- Tetrachoric Correlation.

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## Correlation Formula: TI 83

If you’re taking AP Statistics, you won’t actually have to work the correlation formula by hand. You’ll use your graphing calculator. Here’s how to find r on a TI83.

Step 1: Type your data into a list and make a scatter plot to ensure your variables are roughly correlated. In other words, look for a straight line. Not sure how to do this? See: TI 83 Scatter plot.

Step 2: Press the STAT button.

Step 3: Scroll right to the CALC menu.

Step 4: Scroll down to 4:LinReg(ax+b), then press ENTER. The output will show “r” at the very bottom of the list.

**Tip**: If you don’t see r, turn Diagnostic ON (click here for instructions), then perform the steps again.

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you’re are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.