𝜷 and 𝜸 coefficient of skewness
Karl Pearson defined the following 𝛽 and 𝛾 coefficients of skewness, based upon the central moments of order 2 and 3:
For a symmetrical distribution, 𝛽_1=0.
Tells about the magnitude of the skewness but not its direction.
Karl Pearson’s Gamma coefficient 𝛾_1
𝛾_1=±√(𝛽_1 )=𝑚_3/√((𝑚_2 )^3 )=𝑚_3/(√(𝑚_2 ))^3 =𝑚_3/(√(𝜎^2 ))^3 =𝑚_3/(𝜎)^3
Now the sign of skewness would depend upon the value of 𝑚_3.
Skewness is a measure of how much the distribution is asymmetric.
𝑖^𝑡ℎ frequency density = Frequency per unit interval =𝑓_𝑖/ℎ.
Symmetric distribution: If a perpendicular is drawn on the X-axis, it divides the figure into two congruent parts i.e. identical in all respect or one part can be superimposed on the other i.e., mirror images of each other.
If the left tail and the right tail are equal, we get a symmetric distribution for which mean = median = mode.
Asymmetric distribution: If the right tail is longer, we get a positively skewed distribution for which mean greater than median, which is greater than mode.
If the left tail is longer, we get a negatively skewed distribution for which mean less than median, which is less than mode.
Median is always between Mean and Mode.
In negatively skewed distribution, among these three measures, Mean is the lowest while Mode is the highest value.
In positively skewed distribution, among these three measures, Mode is the lowest while Mean is the highest value.
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Difference between Variance and Skewness
Variance tells us about the amount of variability while skewness gives the direction of variability.
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Where do we need Variance?
In business and economic series, measures of variation have greater practical application than measures of skewness.
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Where do we need Skewness?
In medical and life science field measures of skewness have greater practical applications than the variance.
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Two measures of Skewness
1. Absolute 2. Relative
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Three criteria for a good measure of skewness
1. Should be independent of the underlying unit of the variables so that the symmetry of different distributions with different units can be compared.
2. If the distribution is symmetric, the value of the measure should be zero. Similarly, the measure should give positive or negative values according as the distribution has positive or negative skewness respectively.
3. As we move from extreme negative skewness to extreme positive skewness, the value of the measure should vary accordingly.
00:00 Absolute and relative measures of skewness
01:25 𝜷 and 𝜸 coefficient of skewness
01:47 Formula of 𝜷_1
02:47 Problem of 𝜷_1
04:27 Karl Pearson's Gamma (𝜸) coefficient of skewness
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