## Formula to Calculate Correlation

Correlation is a statistical measure between two variables and is defined as the change of quantity in one variable corresponding to change in another and it is calculated by summation of product of sum of first variable minus the mean of the first variable into sum of second variable minus the mean of second variable divided by whole under root of product of square of the first variable minus mean of first variable into sum of square of second variable minus mean of second variable.

The value of correlation is limited between -1 and +1 and can be interpreted as follows:

**-1:**If it is -1, then variables are known as perfectly negatively correlated. That means if one variable is moving in one direction, then another is moving in the opposite direction.**0:**That means the variable is not having any correlation.**+1:**If it is +1, then variables are known as perfectly positively correlated. Both variables are moving in positive directions.

If we are having 2 variable x and y then correlation coefficient between 2 variables can be found as:

**Correlation Coefficient = ∑(x(i)- mean(x))*(y(i)-mean(y)) / √ (∑(x(i)-mean(x))**

^{2}* ∑(y(i)-mean(y))^{2})Where,

- x(i)= value of x in the sample
- Mean(x) = mean of all values of x
- y(i) = value of y in the sample
- Mean(y) = mean of all values of y

### Examples

It is effortless to calculate the correlation in Excel. Syntax of the function used is as follows:

Correlation Coefficient = CORREL (array1, array2)

#### Example#1

**Let’s take the same example that we have taken above for calculating correlation using excel.**

**Solution:**

Below are the values of x and y:

The calculation is as follows.

Basis excel formula = CORREL (array(x), array(y))

**Coefficient = +0.95**

Since this coefficient is near to +1, hence x and y are highly positively correlated.

#### Example#2** **

**Correlation is mainly useful for analyzing the stock price of companies and creating a stock portfolio based on that.**

**Let us find out the correlation of Apple stock with the Nasdaq index based on the last one-year stock performance. Apple is a US-based multinational company which is specialized in IT products such as iPod, iPad, Mac, etc.**

**Solution:**

Below is the monthly return of Apple and Nasdaq stocks for the last one year:

Let’s now input the values –

Correlation Coefficient = ∑(x(i)- mean(x)).(y(i)-mean(y))/√ ∑(x(i)-mean(x)) ^^{2} ∑(y(i)-mean(y))^^{2}

Correlation between Apple and Nasdaq= 0.039/ (√0.0039)

**Coefficient =0.62**

Since the Correlation between Apple and Nasdaq is positive, hence Apple is positively correlated with Nasdaq.

#### Example#3

**Let us now look upon the correlation between Walmart and Nasdaq index based on the last one-year stock performance. Walmart is a US-based company which is having a retail supermarket chain.**

**Solution:**

Below is the monthly performance between Walmart and Nasdaq for the last one year-

Let’s now input the values in the formula –

Correlation Coefficient = ∑(x(i)- mean(x)).(y(i)-mean(y))/√ ∑(x(i)-mean(x)) ^^{2} ∑(y(i)-mean(y))^^{2}

Therefore, the calculation is as follows,

Correlation between Walmart and Nasdaq= 0.0032/ (√0.0346*0.0219 )

**Coefficient =0.12**

We can see that Walmart and Nasdaq are also positively correlated but not as much compared to Apple correlation with Nasdaq.

### Relevance and Use

A correlation coefficient is useful in establishing the linear relationship between two variables. It measures how a variable will move compared to the movement of another variable. The practical use of this coefficient is to find out the relationship between stock price movement with the overall market movement. Basis of this analysis, a stock analyst, will include the proportion of stocks to create an optimal portfolio with minimum risk. Also, it is useful in data science to find out the relationship between 2 variables.

Also, the correlation coefficient is used very highly for studying the construct validity of data in factor analysis. It is highly used in regression analysis to predict the values of dependent variables based on the relationship between dependent and independent variables. This equation is quite useful in quantitative analysis to get the nature of the relationship between various variables. The basis of this relationship, if a variable is unrelated to other variables, then it can be eliminated from the list.

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