Karl Pearson’s Coefficient of Correlation
What is Karl Pearson’s Coefficient of Correlation?
Karl Pearson’s Coefficient of Correlation, denoted as r, is a measure of the strength and direction of the linear relationship between two continuous variables. It ranges from -1 to +1, where -1 indicates a perfect negative linear correlation, +1 indicates a perfect positive linear correlation, and 0 indicates no linear correlation.
How is Pearson’s Correlation Coefficient calculated?
The formula for calculating Karl Pearson’s Coefficient of Correlation is,
[Tex]Karl~Pearson’s~Coefficient~of~Correlation(r)=\frac{Sum~of~Products~of~Deviations~from~their~respective~means}{Number~of~Pairs\times{Standard~Deviations~of~both~Series}} [/Tex]
Or
[Tex]r=\frac{\sum{xy}}{N\times{\sigma_x}\times{\sigma_y}} [/Tex]
Where,
N = Number of Pair of Observations
x = Deviation of X series from Mean [Tex](X-\bar{X}) [/Tex]
y = Deviation of Y series from Mean [Tex](Y-\bar{Y}) [/Tex]
[Tex]\sigma_x [/Tex] = Standard Deviation of X series [Tex](\sqrt{\frac{\sum{x^2}}{N}}) [/Tex]
[Tex]\sigma_y [/Tex] = Standard Deviation of Y series [Tex](\sqrt{\frac{\sum{y^2}}{N}}) [/Tex]
r = Coefficient of Correlation
What do the values of Pearson’s Correlation Coefficient indicate?
r = +1: Perfect positive linear correlation.
r = −1: Perfect negative linear correlation.
r = 0: No linear correlation.
0 < r < 1: Positive linear correlation of varying strength.
−1 < r < 0: Negative linear correlation of varying strength.
How do you interpret the sign and magnitude of Pearson’s Correlation Coefficient?
Sign: Indicates the direction of the relationship. A positive r means that as one variable increases, the other also increases. A negative r means that as one variable increases, the other decreases.
Magnitude: Indicates the strength of the relationship. Values closer to +1 or -1 signify stronger linear relationships, while values closer to 0 signify weaker linear relationships.
When should Pearson’s Correlation Coefficient not be used?
It should not be used when:
- The relationship between variables is not linear.
- The data contain significant outliers.
- The data are ordinal or categorical.
- The assumptions of homoscedasticity and normality are violated.
How do you interpret a non-significant Pearson’s Correlation Coefficient?
A non-significant r suggests that there is no evidence of a linear relationship between the variables in the sample. This does not necessarily mean there is no relationship at all; the relationship could be non-linear or the sample size could be too small to detect a significant correlation.