Correlation coefficients are used in statistics to measure how strong a relationship is between two variables. There are several types of correlation coefficient. Pearson’s correlation (also called Pearson’s R) is a correlation coefficient commonly used in linear regression. If you’re starting out in statistics, you’ll probably learn about Pearson’s R first. In fact, when anyone refers to the correlation coefficient, they are usually talking about Pearson’s.

## Meaning

- A correlation coefficient of ‘1’ means that for every positive increase in one variable, there is a positive increase of a fixed proportion in the other. For example, shoe sizes go up in (almost) perfect correlation with foot length.
- A correlation coefficient of ‘-1’ means that for every positive increase in one variable, there is a negative decrease of a fixed proportion in the other. For example, the amount of gas in a tank decreases in (almost) perfect correlation with speed.
- 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’.

## Concepts and Application

There are several types of correlation coefficient formulas. One of the most commonly used formulas in stats is Pearson’s correlation is coefficient formula. 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. ‘Sx’ and ‘sy’ are the sample standard deviations, and ‘sxy’ is the sample covariance. The population correlation coefficient uses ‘σx’ and ‘σy’ as the population standard deviations, and ‘σxy’ as the population covariance. Correlation between sets of data is a measure of how well they are related.

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The most common measure of correlation in stats is the Pearson correlation. The full name is the Pearson Product Moment Correlation (PPMC). It shows the linear relationship between two sets of data. In simple terms, it answers the question. The PPMC is not able to tell the difference between dependent variables and independent variables. For example, if you are trying to find the correlation between a high calorie diet and diabetes, you might find a high correlation of 8. However, you could also get the same result with the variables switched around. In other words, you could say that diabetes causes a high calorie diet. That obviously makes no sense. Therefore, as a researcher you have to be aware of the data you are plugging in. In addition, the PPMC will not give you any information about the slope of the line; it only tells you whether there is a relationship.

Pearson correlation is used in thousands of real-life situations. For example, scientists in China wanted to know if there was a relationship between how weedy rice populations are different genetically. The goal was to find out the evolutionary potential of the rice. Pearson’s correlation between the two groups was analyzed. It showed a positive Pearson product moment correlation of between 0.783 and 0.895 for weedy rice populations. This figure is quite high, which suggested a fairly strong relationship.

## Types of Correlations

**Positive and Negative Correlations. **Both the variables (X and Y) will vary in the same direction. If variable X increases, variable Y also will increase; if variable X decreases, variable Y also will decrease. This is positive correlation. If the given variables vary in opposite direction, then they are said to be negatively correlated. If one variable increases, other variable will decrease. In other words, the variables are negatively correlated if there is an inverse relationship between the variables.
**Partial and Multiple Correlations.** In simple correlation, relationships between two variables are studied. In partial and multiple correlations, three or more variables are studied. Three or more variables are simultaneously studied in multiple correlations. In partial correlation more than two variables are studied, but the effect on one variable is kept constant and the relationship between the other two variables is studied.
**Linear and Non-Linear Correlation.** Correlation depends upon the constancy of the ratio of change between the variables. In linear correlation, the percentage change in one variable will be equal to the percentage change in another variable. It is not so in nonlinear correlation.

A correlation coefficient gives you an idea of how well data fits a line or curve. Pearson wasn’t the original inventor of the term correlation but his use of it became one of the most popular ways to measure correlation. Francis Galton (who was also involved with the development of the interquartile range) was the first person to measure correlation, originally termed ‘co-relation’, which actually makes sense considering you’re studying the relationship between a couple of different variables.

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