Correlation Coefficient Formula

Correlation coefficients in mathematics are used to measure the relationship between two variables. However, there are different types of Correlation coefficients but the most common type is Pearson’s correlation that is immensely popular in linear regression.

Most of the times, when we are discussing Correlation coefficients then it is all about Pearson’s until any specific type is not mentioned in the problem. With the help of Correlation coefficients formulas, you can check how strong is the relationship among data. The value is returned between -1 to 1 generally where,

  • One shows the positive relationship or increases in size in the fixed proportion. For example, the size of shoes increases in the perfect correlation with foot length.
  • Minus one shows the negative relationship or decrease in size in the fixed proportion. For example, the amount of gas decreases in perfect correlation with speed.
  • Zero means no relationship at all, or two identities are not related somehow.

\[\large r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}\]


n= Quantity of information.
Σx = Total of the first variable value.
Σy = Total of the second variable value.
Σxy =Sum of the product of first & second value.
Σx2 = Sum of the squares of the first value.
Σy2 = Sum of the squares of the second value.

Linear Correlation Coefficient Formula

Linear correlation is used to find the relationship among two variables in a population. With the help of Formula, you can find how two variables are connected together and the value will always be calculated between -1 and 1. For a strong relationship, the value is 1. If value comes closer to -1 then the relationship is negative and in case of a zero, there is no relationship exists between data given.

Where, n is the number of observations, xi and yi are the variables.

The Coefficient of Variation Formula

To measure the relative variability, the coefficient of variation (CV) formula is used. This is taken as the ratio of standard deviation to the mean. The most common application of CV is the comparison of the result of two surveys or tests. Researchers and scientists are already familiar with the application of CV in different conditions. There could be conducted multiple tests till the time you are not sure of the final outcome.

\[\ Coefficient\;of\;Variation\;Formula = \frac{Standard\;Deviation}{Mean}\]

As per sample and population data type, the formula for standard deviation may vary –

\[\ Sample\;Standard\;Deviation=\frac{\sqrt{\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}}{n-1}\]

\[\ Population\;Standard\;Deviation=\frac{\sqrt{\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}}{n}\]

xi = Terms given in the data
$\overline{x}$ = Mean
n = Total number of terms.

Pearson’s Correlation coefficient formula

There are different states of correlation Formula, one of the most popular types if Pearson’s correlation coefficient formula. For the basic tests, it is used frequently. With the help of Pearson’s Formula, you can quickly identify the dependent or independent variables. For example, how high-calorie diet and diabetes are correlated, researchers generally analyze the associated data before they reach the final conclusion.

\[\large r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}\]

r = Pearson correlation coefficient
x = Values in the first set of data
y = Values in the second set of data
n = Total number of values.

Pearson’s correlation coefficient formula has a multiple of apps in real-life too. For example, if you want to test a portion of population growth for some product then this technique can be used. It is frequently used for a variety of research techniques, sample analysis, and higher studies in colleges. With the basic understanding of the concept, you can apply the Formula for real-life complex problems later during your practical work. If you will check online there are numerous studies where correlation is shown between the two variables. Like, how carbohydrates increased weight and low carbohydrate content in your diet will trigger the weight loss. There are more studies that can be easily found online and used in our daily practices too.

The coefficient of Determination formula

\[\large r = \frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{\left [ n\sum x^{2}-(\sum x)^{2} \right ]\left [ n\sum y^{2}-(\sum y)^{2} \right ]}}\]

r = Correlation coefficient
x = Values in first set of data
y = Values in second set of data
n = Total number of values.