If, instead of having equal probabilities, the values have different probabilities, let x1 have probability p1, x2 have probability p2, ..., xN have probability pN. In this case, the standard deviation will be

Correlation coefficients - measure the strength of association between two variables. The most common correlation coefficient, called the Pearson product-moment correlation coefficient, measures the strength of the linear association between variables.

If we have a series of n measurements of X and Y written as xi and yi where i = 1, 2, ..., n, then the sample correlation coefficient can be used to estimate the population Pearson correlation r between X and Y. The sample correlation coefficient is written

where x and y are the sample means of X and Y, and sx and sy are the sample standard deviations of X and Y.

This can also be written as:

The main result of a correlation is called the correlation coefficient (or "r"). It ranges from -1.0 to +1.0. The closer r is to +1 or -1, the more closely the two variables are related.

If r is close to 0, it means there is no relationship between the variables. If r is positive, it means that as one variable gets larger the other gets larger. If r is negative it means that as one gets larger, the other gets smaller (often called an "inverse" correlation).

Skewness - is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point.

Karl Pearson suggested simpler calculations as a measure of skewness: the Pearson mode or first...