Statistics
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Kurtosis
Kurtosis is the statistical measure that tells us when a distribution is more or less peaked tahn a normal distribution.
• A distribution that is less peaked than normal is called platykurtic.
• A distribution identical to the normal distribution is called mesokurtic.
• A distribution that is more peaked than normal is called leptokurtic.
 platykurtic mesokurtic leptokurtic

Different formulae for measuring kurtosis are:

i) Moment coefficient of kurtosis

One measure of kurtosis uses the fourth moment about the mean expresses in dimensionless form:

which is often denoted as b2

ii) Excess kurtosis:

Kurtosis is more commonly defined as the fourth cumulant divided by the square of the second cumulant, which is equal to the fourth moment around the mean divided by the square of the variance minus 3,

• means leptokurtic distribution.
• means mesokurtic distribution.
•  means platikurtic distribution.

iii) Percentile coefficient of kurtosis

Another measure of kurtosis is based on both quartiles and percentiles and is given by:

where is the semi-interquartile range.

The following frequency distribution indicates the test scores of 40 students in Educational Measurement and Evaluation. Calculate the coefficient of kurtosis.

 C.I. fi 65-69 2 60-64 2 55-59 3 50-54 1 45-49 6 40-44 11 35-39 8 30-34 3 25-29 2 20-24 2

Solution:

 C.I. fi Xi 65-69 2 67 24.125 338742.188 677484.375 60-64 2 62 19.125 133784.492 267568.985 55-59 3 57 14.125 39806.4846 119419.454 50-54 1 52 9.125 6933.16431 6933.16431 45-49 6 47 4.125 289.531494 1737.18896 40-44 11 42 -0.875 0.58618164 6.44799805 35-39 8 37 -5.875 1191.32837 9530.62695 30-34 3 32 -10.875 13986.7581 41960.2742 25-29 2 27 -15.875 63511.8752 127023.75 20-24 2 22 -20.875 189891.68 379783.36 40 16311447.63

Where:

SD = 7.22

(leptokurtic)