Finding the linear correlation coefficient (r) from a table involves using a formula and calculating several summations based on the data provided in the table. Here's a step-by-step guide:
1. Understanding the Formula
The linear correlation coefficient, denoted by 'r', measures the strength and direction of a linear relationship between two variables. The formula to calculate 'r' is:
*r = [n (∑XY) - (∑X)(∑Y)] / √{[n ∑X² - (∑X)²] [n ∑Y² - (∑Y)²]}**
Where:
- r = linear correlation coefficient
- n = number of data points (pairs of x and y values)
- ∑XY = sum of the product of each x and y value
- ∑X = sum of all x values
- ∑Y = sum of all y values
- ∑X² = sum of the squares of all x values
- ∑Y² = sum of the squares of all y values
2. Creating a Table for Calculations
Start with the table of your data, which has columns for 'x' and 'y'. Expand the table to include the following additional columns to facilitate calculations:
| x | y | xy | x² | y² |
|---|---|---|---|---|
| x₁ | y₁ | x₁y₁ | x₁² | y₁² |
| x₂ | y₂ | x₂y₂ | x₂² | y₂² |
| ... | ... | ... | ... | ... |
| xₙ | yₙ | xₙyₙ | xₙ² | yₙ² |
3. Performing the Calculations
- Column 'xy': For each row, multiply the x value by the y value and record the result.
- Column 'x²': For each row, square the x value and record the result.
- Column 'y²': For each row, square the y value and record the result.
4. Summing the Columns
Add up all the values in each column:
- Calculate ∑X (sum of x values).
- Calculate ∑Y (sum of y values).
- Calculate ∑XY (sum of xy values).
- Calculate ∑X² (sum of x² values).
- Calculate ∑Y² (sum of y² values).
5. Plugging the Values into the Formula
Substitute the calculated sums and the number of data points (n) into the correlation coefficient formula:
*r = [n (∑XY) - (∑X)(∑Y)] / √{[n ∑X² - (∑X)²] [n ∑Y² - (∑Y)²]}**
6. Interpreting the Result
The value of 'r' will always be between -1 and +1:
- r = +1: Perfect positive linear correlation.
- r = -1: Perfect negative linear correlation.
- r = 0: No linear correlation.
- Values close to +1 indicate a strong positive correlation.
- Values close to -1 indicate a strong negative correlation.
- Values close to 0 indicate a weak or no linear correlation.
Example
Let's say you have the following data in a table:
| x | y |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 5 |
| 4 | 7 |
| 5 | 9 |
First, create the augmented table:
| x | y | xy | x² | y² |
|---|---|---|---|---|
| 1 | 2 | 2 | 1 | 4 |
| 2 | 4 | 8 | 4 | 16 |
| 3 | 5 | 15 | 9 | 25 |
| 4 | 7 | 28 | 16 | 49 |
| 5 | 9 | 45 | 25 | 81 |
Then, calculate the sums:
- ∑X = 1 + 2 + 3 + 4 + 5 = 15
- ∑Y = 2 + 4 + 5 + 7 + 9 = 27
- ∑XY = 2 + 8 + 15 + 28 + 45 = 98
- ∑X² = 1 + 4 + 9 + 16 + 25 = 55
- ∑Y² = 4 + 16 + 25 + 49 + 81 = 175
- n = 5 (number of data points)
Now, plug these values into the formula:
r = [5(98) - (15)(27)] / √{[5(55) - (15)²] [5(175) - (27)²]}
r = [490 - 405] / √{[275 - 225] [875 - 729]}
r = 85 / √(50 * 146)
r = 85 / √7300
r = 85 / 85.44
r ≈ 0.995
This indicates a very strong positive linear correlation between x and y.
