Calculate The Linear Correlation Coefficient For The Data Below

How to Calculate the Linear Correlation Coefficient (Pearson’s r) – A Step‑by‑Step Guide

When you have two quantitative variables and you want to know how strongly they move together, the linear correlation coefficient—most commonly Pearson’s r—is the go‑to statistic. It tells you both the direction (positive or negative) and the strength of a linear relationship, with values ranging from –1 (perfect negative correlation) through 0 (no linear correlation) to +1 (perfect positive correlation). Below is a complete, easy‑to‑follow walkthrough that includes the formula, the computational steps, an illustrative example, and tips for interpreting the result. Even if you don’t have the exact data set you intended to analyze, the procedure shown here works for any pair of numbers.

1. What Is the Linear Correlation Coefficient?

The linear correlation coefficient (denoted r) quantifies the degree to which two variables, X and Y, vary together in a straight‑line pattern. Mathematically, it is the covariance of the two variables divided by the product of their standard deviations:

[ r ;=; \frac{\displaystyle\sum_{i=1}^{n}(X_i-\bar X)(Y_i-\bar Y)} {\sqrt{\displaystyle\sum_{i=1}^{n}(X_i-\bar X)^2}; \sqrt{\displaystyle\sum_{i=1}^{n}(Y_i-\bar Y)^2}} ]

where

• (n) = number of paired observations,
• (\bar X) and (\bar Y) = sample means of X and Y, * (X_i) and (Y_i) = individual data points.

Key properties * Scale‑free – r does not depend on the units of measurement.

• Symmetrical – swapping X and Y leaves r unchanged.
• Sensitive to outliers – a single extreme point can dramatically alter the value.

2. Step‑by‑Step Calculation Procedure

Below is a practical checklist you can follow with any data set. Each step builds on the previous one, making it easy to spot mistakes.

Tip: Using a spreadsheet (Excel, Google Sheets) or a statistical calculator automates steps 3‑6, but doing them by hand at least once reinforces understanding.

3. Worked Example (Illustrative Data Set)

Since the original prompt did not include a specific data table, we will demonstrate the calculation with a small, easy‑to‑follow dataset. Feel free to replace the numbers with your own data; the mechanics stay identical.

Step 1–2: Compute the Means

[ \bar X = \frac{2+3+5+6+8}{5} = \frac{24}{5} = 4.8 ] [ \bar Y = \frac{65+70+78+82+88}{5} = \frac{383}{5} = 76.6 ]

Step 3–6: Build the Table of Deviations, Cross‑Products, and Squares