Pearson Correlation Coefficient

Pearson Correlation Coefficient: Complete Guide with Examples

What is Pearson Correlation Coefficient?
Pearson Correlation Formula
When to Use Pearson Correlation?
When Not to Use Pearson Correlation?
Step-by-Step Calculation Example
Key Takeaways

What is Pearson Correlation Coefficient?

The Pearson Correlation Coefficient (denoted as r) is a statistical measure that quantifies the strength and direction of the linear relationship between two variables (X and Y). Its value ranges between -1 and +1:

-1

Perfect Negative

-0.5

Weak Negative

No Correlation

+0.5

Weak Positive

Perfect Positive

+1: Perfect positive linear relationship (as X increases, Y increases)
-1: Perfect negative linear relationship (as X increases, Y decreases)
0: No linear relationship (no consistent pattern between X and Y)

Pearson Correlation 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:

xᵢ and yᵢ: Individual data points for X and Y
x̄ and ȳ: Means (averages) of X and Y
Numerator: Covariance between X and Y
Denominator: Product of standard deviations of X and Y

When to Use Pearson Correlation?

Situation	Description	Example
Linear Relationship Assessment	When checking for linear relationship between two variables	Height vs. Weight, Study hours vs. Exam scores
Continuous Variables	When both variables are continuous	Temperature, Time, Age, Income
Normally Distributed Data	When data follows approximately normal distribution	Standardized test scores, Biological measurements
Research and Analysis	In scientific studies to understand relationships	Economic indicators, Psychological studies

Examples of Appropriate Use:

Relationship between study hours and exam scores
Correlation between temperature and ice cream sales
Connection between advertising expenditure and sales revenue

When Not to Use Pearson Correlation?

Pearson Correlation should not be used when:

Situation	Problem	Alternative Approach
Non-Linear Relationships	Only measures linear relationships	Spearman’s rank or non-linear correlation
Categorical Data	Requires continuous variables	Chi-square test or other categorical methods
Presence of Outliers	Outliers can distort results	Robust correlation measures
Non-Normal Data	Assumes normal distribution	Non-parametric methods
Assuming Causation	Correlation ≠ Causation	Experimental design for causation

Example of Inappropriate Use:

Checking correlation between “ice cream sales” and “shark attacks” can be misleading because both increase in summer (seasonal effect), but there’s no direct causation.

Step-by-Step Calculation Example

Problem Statement:

Calculate the Pearson Correlation Coefficient for the following dataset of 5 students’ study hours (X) and exam scores (Y):

Student	Study Hours (X)	Exam Scores (Y)
1	2	50
2	4	60
3	6	75
4	8	85
5	10	90

Step 1: Calculate Means

\[ \bar{x} = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6 \] \[ \bar{y} = \frac{50 + 60 + 75 + 85 + 90}{5} = \frac{360}{5} = 72 \]

Step 2: Calculate Deviations and Products

Create a table for (xᵢ – x̄), (yᵢ – ȳ), and their products:

xᵢ	yᵢ	xᵢ – x̄	yᵢ – ȳ	(xᵢ – x̄)(yᵢ – ȳ)	(xᵢ – x̄)²	(yᵢ – ȳ)²
2	50	-4	-22	88	16	484
4	60	-2	-12	24	4	144
6	75	0	3	0	0	9
8	85	2	13	26	4	169
10	90	4	18	72	16	324

Step 3: Sum the Columns

\[ \sum(x_i – \bar{x})(y_i – \bar{y}) = 88 + 24 + 0 + 26 + 72 = 210 \] \[ \sum(x_i – \bar{x})^2 = 16 + 4 + 0 + 4 + 16 = 40 \] \[ \sum(y_i – \bar{y})^2 = 484 + 144 + 9 + 169 + 324 = 1130 \]

Step 4: Apply the Formula

\[ r = \frac{210}{\sqrt{40 \times 1130}} \] \[ r = \frac{210}{\sqrt{45200}} \] \[ r = \frac{210}{212.62} \approx 0.987 \]

Step 5: Interpretation

r = 0.987 indicates a very strong positive linear relationship between study hours and exam scores.

This means that as study hours increase, exam scores also consistently increase.

Key Takeaways

Pearson Correlation (r) measures linear relationship between two continuous variables (-1 to +1)
Use for normally distributed continuous data with linear relationships
Avoid for non-linear data, categorical variables, or when outliers are present
Correlation does not imply causation
In our example, r = 0.987 showed very strong positive correlation between study hours and exam scores

Tax Calculater