Moments. Skewness. Kurtosis.

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Moments

r^th moment. If x₁, x₂, .... , x_nare the n values assumed by the variable x, the quantity

is called the r^th moment. The first moment with r = 1 corresponds to the arithmetic mean.

r^th moment about the mean. If x₁, x₂, .... , x_nare the n values assumed by the variable x, the quantity

is called the r^th moment about the mean . If r = 1, m₁ = 0 (the first moment about the mean is 0). If r = 2, m₂ = s², the variance.

r^th moment about any origin A. If x₁, x₂, .... , x_nare the n values assumed by the variable x, the quantity

where d = x - A are the deviations of x from A. If A = 0, 3) reduces to 1). Consequently 1) is often called the r^th moment about zero.

Moments for grouped data

If x₁, x₂, .... , x_n occur with frequencies f₁, f₂. ... , f_k respectively, the above moments are given by

r^th moment.

r^th moment about the mean.

r^th moment about any origin A.

Relationships between moments

The following relationships exist between moments about the mean m_r and moments about an arbitrary origin m_r'.

m₂ = m₂' - m₁'²

7) m₃ = m₃' - 3m₁'m₂' + 2m₁'³

m₄ = m₄'- 4m₁'m₃' + 6m₁'²m₂' - 3m₁'⁴

Note that m₁' = - A.

Computations of moments for grouped data

The coding method used for computing the mean and standard deviation can also be used to provide a short method for calculating moments. The method uses the fact that x_i = A + cu_i so that from equation 6) we have

which can be used to find m_r using equations 7).

Moments in dimensionless form

To avoid particular units we can define the dimensionless moments about the mean

where s = is the standard deviation.

Since m_{1 =} 0 and m₂ = s², we have a₁ = 0, a₂ =1.

Skewness

Skewness is the degree of asymmetry, or departure from symmetry, of a distribution. If the frequency curve (smoothed frequency polygon) of a distribution has a longer “tail” to the right of the central maximum than to the left, the distribution is said to be skewed to the right or to have positive skewness. If it has a longer tail to the left of the central maximum, it is said to be skewed to the left or have negative skewness.

For skewed distributions the mean tends to lie on the same side of the mode as the longer tail. Thus a measure of asymmetry is supplied by the difference (Mean - Mode). This can be made dimensionless on division by a measure of dispersion, such as the standard deviation, leading to the definition

If we wish to avoid use of the mode we can employ the empirical formula

Mean - Mode = 3(Mean - Median)

giving

These two measures are called, respectively, Pearson’s first and second coefficients of skewness.

Other measures of skewness are defined in terms of quartiles and percentiles as follows:

Another important measure of skewness uses the third moment about the mean expressed in dimensionless form:

Still another measure of skewness is sometimes given by

15) b₁ = a₃²

For perfectly symmetrical curves, such as the normal curve, a₃ and b₁ are zero.

Kurtosis

Kurtosis is the degree of peakedness of a distribution, usually taken relative to a normal distribution. A distribution having a relatively high peak such as the curve of Fig. 1 (a) is called leptokurtic, while a curve that is flat-topped is called platykurtic. The normal distribution is called mesokurtic.

One measure of kurtosis uses the fourth moment about the mean expressed in dimensionless form and is given by

16) Moment coefficient of kurtosis = a₄ = m₄/s₄ = m₄/m₂²

and often designated by b₂.

Another measure of kurtosis that is also used is the percentile coefficient of kurtosis based on both quartiles and percentiles and given by

17) κ = Q / (P₉₀ - P₁₀)

where Q = ½(Q₃ - Q₁) is the semi-interquartile range. For normal distributions it has a value of 0.263.

References

Murray R Spiegel. Statistics (Schaum Publishing Co.)