Correlation Coefficient 

In the world of statistics, correlation coefficient is an essential measure used to describe the relationship between two variables. Whether you’re a student of statistics, a data analyst, or a researcher, understanding the correlation coefficient can help you gain insights into how variables interact. In this section, we’ll explore what the correlation coefficient is, how it’s calculated, its types, and how to interpret it.

1.0What is Correlation Coefficient?

A correlation coefficient is a statistical measure that describes the strength and direction of the relationship between two variables. It tells us whether the variables tend to increase or decrease together and the degree of this association. The correlation coefficient can vary from -1 to +1:

• +1 signifies a perfect positive correlation (as one variable increases, the other increases proportionally).
• -1 signifies a perfect negative correlation (as one variable increases, the other decreases proportionally).
• 0 signifies no linear relationship between the variables.

Pearson's correlation coefficient is the most widely used measure of correlation, but there are others, such as Spearman’s rank correlation, which we will also discuss.

2.0Calculating Pearson’s Correlation Coefficient

Pearson’s correlation coefficient (r) is the most widely used measure of correlation. It is calculated using the following formula:

$r=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x}{)}^2\sum (y_i-\bar{y}{)}^2}}$

Where:

• $x_i$and $y_i$ are the individual data points in the datasets X and Y,
• $\bar{x}$ and $\bar{y}$ are the means of the datasets X and Y.

This formula computes the covariance of the variables divided by the product of their standard deviations. The result, r, tells you how strongly the variables X and Y are related.

3.0Correlation Coefficient and Coefficient of Determination

The coefficient of determination, denoted as $r^2$, is closely related to the correlation coefficient. It represents the proportion of the variance in one variable that can be explained by the other variable. If r = 0.8, then $r^2=0.64$, meaning that 64% of the variance in one variable can be explained by the other. The coefficient of determination is particularly useful when analyzing linear regression models.

4.0Interpreting The Correlation Coefficient

The value of the correlation coefficient indicates the strength and direction of the relationship:

• Positive correlation (r > 0): As one variable increases, the other variable also increases.
• Negative correlation (r < 0): As one variable increases, the other decreases.
• No correlation (r = 0): There is no linear relationship between the variables.

In general:

• 0.1 to 0.3: Weak positive correlation.
• 0.3 to 0.5: Moderate positive correlation.
• 0.5 to 1: Strong positive correlation.
• -0.1 to -0.3: Weak negative correlation.
• -0.3 to -0.5: Moderate negative correlation.
• -0.5 to -1: Strong negative correlation.

5.0Correlation Coefficient Example

Let’s say you want to determine the correlation coefficient between the number of hours studied (X) and the scores on a test (Y). Using the data points, you can calculate the Pearson’s correlation coefficient to see if there’s a strong or weak relationship between these two variables.

Example data:

X = [1, 2, 3, 4, 5], Y = [2, 4, 5, 4, 5] 

Calculating the correlation coefficient for this data would give you a numerical value that tells you the degree to which hours studied is related to test scores.

6.0Formula for Spearman’s Rank Correlation Coefficient

While Pearson’s correlation measures linear relationships, Spearman’s rank correlation coefficient (ρ) measures the strength and direction of a monotonic relationship between two variables. It is used when the data is not linear or when the variables are ordinal (ranked).

The formula for Spearman’s rank correlation is:

$\rho =1-\frac{6\sum d_1^2}{n(n^2-1)}$

Where:

• $d_i$ is the difference between the ranks of corresponding values in the two variables,
• n is the number of data points.

Spearman’s rank correlation coefficient also ranges from –1 to +1, with the same interpretation as Pearson’s.

7.0Types of Correlation Coefficients

There are several types of correlation coefficients, depending on the nature of the data:

1. Pearson’s correlation coefficient: Used for measuring the linear relationship between two continuous variables.
2. Spearman’s rank correlation coefficient: Used for ordinal data or when the relationship is not linear but monotonic.
3. Kendall’s tau: Another non-parametric measure of association, commonly used for ordinal data.

8.0Determining Correlation Coefficient

To determine the correlation coefficient:

1. Choose the type of correlation (Pearson, Spearman, Kendall, etc.).
2. Collect the data points for the two variables.
3. Use the appropriate formula to calculate the correlation coefficient.
4. Interpret the result to understand the strength and direction of the relationship.

9.0Solved Example On Correlation Coefficient

Example 1: Given the following data for two variables, X and Y, calculate the Pearson correlation coefficient.

Solution: 

Step 1: Calculate the necessary sums.

• $\sum X=1+2+3+4+5=15$
• $\sum Y=2+3+5+7+8=25$
• $\sum XY=(1\ast 2)+(2\ast 3)+(3\ast 5)+(4\ast 7)+(5\ast 8)=2+6+15+28+40=91$
• $\sum X^2=1^2+2^2+3^2+4^2+5^2=1+4+9+16+25=55$
• $\sum Y^2=2^2+3^2+5^2+7^2+8^2=4+9+25+49+64=151$

Step 2: Apply the Pearson formula.

The formula for the Pearson correlation coefficient is:

$r=\frac{n(\sum XY)-(\sum X)(\sum Y)}{\sqrt{[n\sum X^2-(\sum X{)}^2][n\sum Y^2-(\sum Y{)}^2]}}$

Where:

• n = 5 (the number of data points)
• $\sum X=15$
• $\sum Y=25$
• $\sum XY=91$
• $\sum X^2=55$
• $\sum Y^2=151$

Step 3: Plug in the values into the formula.

$r=\frac{5(91)-(15)(25)}{\sqrt{[5(55)-(15{)}^2][5(151)-(25{)}^2]}}$

Simplify the numerator:

$r=\frac{455-375}{\sqrt{[275-225][755-625]}}$

$r=\frac{80}{\sqrt{[50][130]}}$

$r=\frac{80}{\sqrt{6500}}$

$r=\frac{80}{80.62}$

$r\approx 0.994$

Step 4: Interpretation

The correlation coefficient r ≈ 0.994 indicates a very strong positive linear relationship between variables X and Y.

Example 2: Calculate the correlation coefficient for the following height (in inches) of the father (x) and their son (y).

We know that $r=\frac{\text{Cov}(x,y)}{{\sigma }_x{\sigma }_y}$

$\text{Cov}(x,y)=\frac{\sum xy}{n}-\left(\frac{\sum x}{n}\times \frac{\sum y}{n}\right)$

$=\frac{37560}{8}-\left(\frac{544}{8}\times \frac{552}{8}\right)$

= 4635 – 4692 = 3

${\sigma }_x=\sqrt{\frac{\sum x^2}{n}-{\left(\frac{\sum x}{n}\right)}^2}$

$=\sqrt{\frac{37028}{8}-4624}$

= 2.121

${\sigma }_y=\sqrt{\frac{\sum y^2}{n}-{\left(\frac{\sum y}{n}\right)}^2}$

$=\sqrt{\frac{38131}{8}-{\left(\frac{552}{8}\right)}^2}$

= 2.345

Then $r=\frac{3}{(2.121)(2.345)}=0.6032$