Population Correlation Coefficient

A Population Correlation Coefficient is a correlation coefficient (between two random variables) that is a population statistic.

• Context:
  • It can be defined as [math]\displaystyle{ \rho(x,y) = \frac{1}{N}\sum_{i=1}^N \frac{(x_i-\mu_x)(y_i-\mu_y)}{\sigma_x \sigma_y}=\frac{cov(x,y)}{\sigma_x \sigma_y} }[/math] where [math]\displaystyle{ \mu_x }[/math], [math]\displaystyle{ \mu_y }[/math], [math]\displaystyle{ \sigma_x }[/math] and [math]\displaystyle{ \sigma_y }[/math] are the population mean values and population standard deviations of the random variables [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], [math]\displaystyle{ {cov(x,y)} }[/math] is the population covariance value.
• Example(s)
  • a Pearson Product-Moment Correlation Coefficient.
  • …
• Counter-Example(s)
  • Sample Correlation Coefficient.
  • Population Variance Value.
  • Population Covariance Value.
• See: Pearson Correlation Test, Correlation Coefficient, Linear Regression, Co-Occurrence Matrix.

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References

2017

• (Stattrek,2017) rArr; http://stattrek.com/statistics/correlation.aspx
  • (...) ::: Population correlation coefficient. The correlation ρ between two variables is:

[math]\displaystyle{ ρ = [ 1 / N ] * Σ { [ (X_i - μ_X) / σ_x ] * [ (Y_i - μ_Y) / σ_y ] } }[/math]

where N is the number of observations in the population, Σ is the summation symbol, [math]\displaystyle{ X_i }[/math] is the X value for observation i, μX is the population mean for variable X, [math]\displaystyle{ Y_i }[/math] is the Y value for observation i, [math]\displaystyle{ μ_Y }[/math] is the population mean for variable Y, [math]\displaystyle{ σ_x }[/math] is the population standard deviation of X, and σy is the population standard deviation of Y.

The formula below uses sample means and sample standard deviations to compute a correlation coefficient (r) from sample data.

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient Retrieved:2015-2-8.
  • In statistics, the Pearson product-moment correlation coefficient () (sometimes referred to as the PPMCC or PCC or Pearson's r) is a measure of the [[linear

correlation]] (dependence) between two variables X and Y, giving a value between +1 and −1 inclusive, where 1 is total positive correlation, 0 is no correlation, and −1 is total negative correlation. It is widely used in the sciences as a measure of the degree of linear dependence between two variables. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s.

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