Calculate the Pearson correlation coefficient?

Pearson Correlation Coefficient

The Pearson correlation coefficient, also known as the Pearson product-moment correlation coefficient, is a measure of the linear relationship between two quantitative variables. It ranges from -1 to 1, where:

- -1: Perfect negative correlation
- 0: No correlation
- 1: Perfect positive correlation

Formula:

```
r = (Cov(X,Y)) / (SD(X) - SD(Y))
```

where:

- r: Pearson correlation coefficient
- Cov(X,Y): Covariance between variables X and Y
- SD(X): Standard deviation of variable X
- SD(Y): Standard deviation of variable Y

Steps to Calculate:

1. Calculate the covariance using the formula:
```
Cov(X,Y) = Σ((X - μX) - (Y - μY)) / (n - 1)
```
where:
- μX: Mean of variable X
- μY: Mean of variable Y
- n: Number of data points

2. Calculate the standard deviations using the formula:
```
SD(X) = √(Σ((X - μX)^2) / (n - 1))
SD(Y) = √(Σ((Y - μY)^2) / (n - 1))
```

3. Substitute the values of covariance and standard deviations into the Pearson correlation coefficient formula:
```
r = (Cov(X,Y)) / (SD(X) - SD(Y))
```

Example:

Consider the following data points:

```
(X,Y) = {(1,2), (3,5), (5,8), (7,11)}
```

1. Calculate the covariance:
```
Cov(X,Y) = ((1-4)-(2-7) + (3-4)-(5-7) + (5-4)-(8-7) + (7-4)-(11-7)) / (4-1) = 3
```

2. Calculate the standard deviations:
```
SD(X) = √(((1-4)^2 + (3-4)^2 + (5-4)^2 + (7-4)^2) / (4-1)) = 2.24
SD(Y) = √(((2-7)^2 + (5-7)^2 + (8-7)^2 + (11-7)^2) / (4-1)) = 3.61
```

3. Calculate the Pearson correlation coefficient:
```
r = (3) / (2.24 - 3.61) = 0.8
```

Therefore, there is a strong positive correlation between variables X and Y, with a Pearson correlation coefficient of 0.8.