when data are skewed to the left the measure of skewness is more likely to be negative skew.
when data are skewed to the left the measure of skewness is more likely to be negative skew.
when data are skewed to the left the measure of skewness is more likely to be?
negative skew
the third central moment is the measure of the lopsidedness of the distribution; any symmetric distribution will have a third central moment, if defined, of zero. the normalised third central moment is called the skewness, often . a distribution that is skewed to the left (the tail of the distribution is longer on the left) will have a negative skewness. a distribution that is skewed to the right (the tail of the distribution is longer on the right), will have a positive skewness.
moment (mathematics)
the accompanying plot shows these two solutions as surfaces in a space with horizontal axes of (sample excess kurtosis) and (sample squared skewness) and the shape parameters as the vertical axis. the surfaces are constrained by the condition that the sample excess kurtosis must be bounded by the sample squared skewness as stipulated in the above equation. the two surfaces meet at the right edge defined by zero skewness. along this right edge, both parameters are equal and the distribution is symmetric u-shaped for α = β  1, uniform for α = β = 1, upside-down-u-shaped for 1  α = β  2 and bell-shaped for α = β  2. the surfaces also meet at the front (lower) edge defined by "the impossible boundary" line (excess kurtosis + 2 - skewness = 0). along this front (lower) boundary both shape parameters approach zero, and the probability density is concentrated more at one end than the other end (with practically nothing in between), with probabilities formula_216 at the left end "x" = 0 and formula_217 at the right end "x" = 1. the two surfaces become further apart towards the rear edge. at this rear edge the surface parameters are quite different from each other. as remarked, for example, by bowman and shenton, sampling in the neighborhood of the line (sample excess kurtosis - (3/2)(sample skewness) = 0) (the just-j-shaped portion of the rear edge where blue meets beige), "is dangerously near to chaos", because at that line the denominator of the expression above for the estimate ν = α + β becomes zero and hence ν approaches infinity as that line is approached. bowman and shenton write that "the higher moment parameters (kurtosis and skewness) are extremely fragile (near that line). however the mean and standard deviation are fairly reliable." therefore, the problem is for the case of four parameter estimation for very skewed distributions such that the excess kurtosis approaches (3/2) times the square of the skewness. this boundary line is produced by extremely skewed distributions with very large values of one of the parameters and very small values of the other parameter. see section titled "kurtosis bounded by the square of the skewness" for a numerical example and further comments about this rear edge boundary line (sample excess kurtosis - (3/2)(sample skewness) = 0). as remarked by karl pearson himself this issue may not be of much practical importance as this trouble arises only for very skewed j-shaped (or mirror-image j-shaped) distributions with very different values of shape parameters that are unlikely to occur much in practice). the usual skewed-bell-shape distributions that occur in practice do not have this parameter estimation problem.
beta distribution