Statistical Parameters of a Distribution The mean of X is a measure of the center of the distribution of X The variance of X is the second moment about the mean, and measures the spread of the distribution of X about the mean The third and fourth moments of X about the mean also measure interesting features of the distribution The third moment measures skewness, the lack of symmetry of the distribution The fourth moment measures kurtosis, the degree to which the distribution is peaked The actual numerical measures of these characteristics are standardized to eliminate the physical units, by dividing by an appropriate power of the standard deviation. Statistical Parameters Measure of Skewness The distribution X is said to be positively skewed, negatively skewed or unskewed depending on whether skew(X) is positive, negative, or 0. Skewness: indicator used in distribution analysis as a sign of asymmetry and deviation from a normal distribution. In the unimodal case, Skewness > 0 - Right skewed distribution - most values are concentrated on left of the mean, with extreme values to the right. Skewness < 0 - Left skewed distribution - most values are concentrated on the right of the mean, with extreme values to the left. Skewness = 0 - mean = median, the distribution is symmetrical around the mean. E[( x x )3 ] x 3 , Measure of Skewness Relationships of the Mean, Median, and Mode in Symmetrical and Skewed Distributions Skewness and Kurtosis Measure of Kurtosis Kurtosis is always nonnegative, and in fact is strictly positive unless X is a constant Kurtosis - indicator used in distribution analysis as a sign of flattening or "peakedness" of a distribution. E[( x x ) 4 ] . 4 In the unimodal case, Kurtosis > 3 - Leptokurtic distribution, sharper than a normal distribution, with values concentrated around the mean and thicker tails. This means high probability for extreme values. Kurtosis < 3 - Platykurtic distribution, flatter than a normal distribution with a wider peak. The probability for extreme values is less than for a normal distribution, and the values are wider spread around the mean. Kurtosis = 3 - Mesokurtic distribution - normal distribution for example. The kurtosis of the standard normal distribution is 3. Using the standard normal distribution as a benchmark, the excess kurtosis of a random variable X is defined to be kurt(X)−3. It is defined as excess kurtosis Rayleigh Distribution Assuming that each component is uncorrelated, normally distributed with equal variance, and zero mean, then the overall wind speed (vector magnitude) will be characterized by a Rayleigh distribution. Noise Probability Density Functions PDF’s of noise and signal plus noise Normal (or Gaussian) distribution