Basic concepts of probability and statistics

Distribution moments

Some characteristics of probability distributions are described by their moments, either about the origin or about the mean.

For a random variable X with probability density function f(x), the rth moment about the origin is defined as the expected value of rth power of X:


The rth moment about the mean, or central moment, is defined with:


The first moment about the origin is the mean m of the variable X:


The mean describes the location of the distribution; it is a measure of central tendency. Geometrically, the mean defines the center of gravity of the p.d.f. The second moment about the mean is the variance, denoted either var[X] or s2:


The standard deviation s is the square root of the variance and describes variability or spreading of the values around the mean; it as a measure of scale. Larger s means wider spread of data around the mean, and vice versa. A dimensionless measure of variability is the coefficient of variation, Cv, defined as the ratio between the standard deviation and the mean:

The third moment about the mean describes the symmetry, or the skewness, of the values in the distribution:

The coefficient of skewness is the usual dimensionless measure of symmetry:

The skewness (m3 and Cs) will be positive if positive differences (X m) prevail over negative differences; this means that the right tail of p.d.f. is longer then the left tail, i.e. the distribution is skewed to the right. Negative skewness implies that the distribution is skewed to the left.

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