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Correlation: Formula & Types

Instructor: Sharon Linde
Imagine you have two data sets and you want to know how closely the two variables are related to each other. Correlation is a tool that allows you to do this. This lesson describes how to calculate correlation and interpret the results. Intrigued? Read on for details.

Definition of Correlation

In normal everyday language a correlation implies a relationship between two or more things. You may correlate the smell of crayons to your youth, or the sound of waves to vacation. In mathematics correlation is a measurement of the covariance or dependence of one variable on another. Basically it's asking the question, 'If I increase this variable by one unit, how well can I predict what will happen in the other variable?' Good question, right? Let's see how this works.

Types of Correlations

Broadly speaking there are three different types of correlations: positive, negative, and neutral or no correlation. A perfect positive correlation would mean that if you increased the one variable by one unit you could predict with 100% accuracy how far the other variable would increase. A perfect negative correlation would indicate a 100% accurate prediction of the decrease in the other variable. If there is no correlation, or a neutral correlation between two variables then changing one variable will have no predictable result on the other variable.

Correlation
correlation examples

Of course, there are going to be relationships that fall on the spectrum between these three general descriptions. That's where formulas come in to play.

Formulas for Correlations

The full formula for correlation is quite long - take a look:

Correlation Calculation
correlation

As you can see, the above equation can get quite time consuming for even just a few data points. This is why this calculation is almost always done by computers.

Calculations - check! Let's see how to interpret the correlation number once you have it.

Range and Interpretation of the Correlation Coefficient

So you've done the hard part of calculating the correlation coefficient (r) - now what? The more important part is answering the question, 'What does this number mean?'

Whichever method you use to calculate the correlation coefficient you will get a number between -1.0 and +1.0, inclusive. A result of 1.0 would indicate a perfect positive correlation, 0.00 gives no indication of correlation, and -1.0 is a perfect negative correlation. Anything in between zero and one would indicate a less than perfect positive correlation and anything between negative one and zero a less than perfect negative correlation.

Let's say you calculated a correlation of 0.95 between two data sets. Does this mean that for every unit you increase the one variable by, you will increase the other one by 0.95 units? While this may be a common assumption it turns out to be mistaken. If you took 10 points on the line y = 0.01x and calculated the correlation, you would get 1.0. The same is true for points on the line y = 100x. A high correlation between the two sets does not give you any information about how much one variable will respond to changes in the other.

Perfect Correlation
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Another way to think about this is in terms of regression lines. The regression line gives you a prediction of one variable when you change the other but correlation tells you how accurate that prediction is likely to be.

It is also very important to note that correlation and causation are not the same thing. Two variables having a correlation, even a very strong correlation, does NOT mean that one of them causes the other.

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