notes on measures of skewness and kurtosis
these notes are introduction to measures of moments, skewness and kurtosis. these notes are help full for students studying subjects like statistics , or doing economics hons. from ignou or d.u , beside this any one interested in learning about measures of moment, skewness and kurtosis can refer to these notes.
Given n observations i = 1, 2, 3, …..n and an arbitrary constant A for
ungrouped frequency distribution
we denote
1st order moment around any constant A as m1
2nd order as m2
and so on for 3rd,4th..... moment around any arbitrary constant
for grouped frequency distribution
_
if A = x we call them central moments and denote them by μ1, μ2, μ3 respectively.
μ1 = 0 (first order central moment)
μ2 (second order central moment)= Var (x).
measuring Skewness
The figures below explain the three types of skewness and their properties in terms of mean, median and mode.
Mn implies mean, Md implies median and Mo implies mode.
The frequency distribution of a variable is said to be symmetric if the
frequencies of the variable are symmetrically distributed in both sides of the
mean.Symmetric distributions are generally bell shaped and mean, median and
mode of these distributions coincide.
methods by which we can measure skewness of a distribution.
if (mean – median) is positive the distribution is positively skewed and if it is negative
the distribution is negatively skewed.
Pearson’s Measures -
The more the median and mean are distant the more skew a distribution is. Pearson takes this property of a distribution to derive a measure of skewness.
Pearsonian first measure = (Mean – Mode) / Standard Deviation.
Pearsonian Second Measure = 3 (Mean – Median)/ Standard Deviation
The measures are relative to S.D. to make them unit free.
Moment Measure:
Bowley’s Measures: Bowley’s measure of skewness is given by the following formula:
For an exactly symmetrical distribution, Q2 ( Median ) lies exactly between Q1
and Q2 . For a positively skewed distribution i.e., when the longest tail of the
frequency lies to the right, Q3 will be wider away from Q2 than Q1 and vice
versa for negatively skewed distribution. The arithmetic mean of the
difference between of Q1 and Q3 from Q2 (median) taken relative to quartile deviation gives Bowley’s measure of skewness.
kurtosis
Kurtosis refers to the degree of peakedness of the frequency curve.The
more dense the observations near the mode, the sharper is the peak of the
frequency distribution. This characteristic of frequency distribution is known
as kurtosis.
The only measure of kurtosis is based on moments.
Kurtosis (γ2) = ( μ4 /σ 4 ) - 3
where μ4 is the fourth order central moment.
Kurtosis of a distribution could be of three types.
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Given n observations i = 1, 2, 3, …..n and an arbitrary constant A for
ungrouped frequency distribution
we denote
1st order moment around any constant A as m1
2nd order as m2
and so on for 3rd,4th..... moment around any arbitrary constant
for grouped frequency distribution
_
if A = x we call them central moments and denote them by μ1, μ2, μ3 respectively.
μ1 = 0 (first order central moment)
μ2 (second order central moment)= Var (x).
measuring Skewness
The figures below explain the three types of skewness and their properties in terms of mean, median and mode.
Mn implies mean, Md implies median and Mo implies mode.
The frequency distribution of a variable is said to be symmetric if the
frequencies of the variable are symmetrically distributed in both sides of the
mean.Symmetric distributions are generally bell shaped and mean, median and
mode of these distributions coincide.
methods by which we can measure skewness of a distribution.
if (mean – median) is positive the distribution is positively skewed and if it is negative
the distribution is negatively skewed.
Pearson’s Measures -
The more the median and mean are distant the more skew a distribution is. Pearson takes this property of a distribution to derive a measure of skewness.
Pearsonian first measure = (Mean – Mode) / Standard Deviation.
Pearsonian Second Measure = 3 (Mean – Median)/ Standard Deviation
The measures are relative to S.D. to make them unit free.
Moment Measure:
Bowley’s Measures: Bowley’s measure of skewness is given by the following formula:
For an exactly symmetrical distribution, Q2 ( Median ) lies exactly between Q1
and Q2 . For a positively skewed distribution i.e., when the longest tail of the
frequency lies to the right, Q3 will be wider away from Q2 than Q1 and vice
versa for negatively skewed distribution. The arithmetic mean of the
difference between of Q1 and Q3 from Q2 (median) taken relative to quartile deviation gives Bowley’s measure of skewness.
kurtosis
Kurtosis refers to the degree of peakedness of the frequency curve.The
more dense the observations near the mode, the sharper is the peak of the
frequency distribution. This characteristic of frequency distribution is known
as kurtosis.
The only measure of kurtosis is based on moments.
Kurtosis (γ2) = ( μ4 /σ 4 ) - 3
where μ4 is the fourth order central moment.
Kurtosis of a distribution could be of three types.
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