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Examples of Kurtosis Calculation (Step by Step)

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The Kurtosis of a distribution gives us some idea about the shape of the distribution. Depending on the value of the Kurtosis we can classify our data as leptokurtic, mesokurtic, and platykurtic. The Kurtosis of a given set of ungrouped data values can be calculated using the formula, \text{Kurtosis }= \frac{\sum (x_i-\bar{x})^4}{n\sigma^4} where, \bar{x} denotes the mean and \sigma denotes the standard deviation.

Example 1: Kurtosis for Ungrouped Data

Consider the following set of ungrouped data values, 23, 34, 38, 47, 59, 63, 84. We first calculate the values of the mean and the standard deviation. \bar{x}=\frac{\sum x_i}{n} = \frac{23+34+38+47+59+63+84}{7}= \frac{348}{7} = 49.7143.

x_i(x_i-\bar{x})(x_i-\bar{x})^2(x_i-\bar{x})^3(x_i-\bar{x})^4
=(x_i-49.7143)=(x_i-49.7143)^2=(x_i-49.7143)^3=(x_i-49.7143)^4
23-26.714713.653-19065509301
34-15.714246.939-3880.560978.8
38-11.714137.225-1607.518830.6
47-2.71437.3673-19.99754.2778
599.285786.2245800.6567434.66
6313.2857176.512345.0631155.9
8434.28571175.5140303.21381824
TOTAL = 348TOTAL = 0TOTAL = 2543.43TOTAL = 18876.2TOTAL = 2009579
\text{Standard Deviation }\sigma =\sqrt{\frac{\sum (x_i-\bar{x})^2}{n}} = \sqrt{\frac{2543.3}{7}}= 19.0617

We now calculate the Kurtosis using the formula,\text{Kurtosis }= \frac{\sum (x_i-\bar{x})^4}{n\sigma^4} \text{Kurtosis }= \frac{2009579}{7\times 19.0617^4} = 2.1745

Example 2: Grouped Data Kurtosis Calculation

Consider the following set of data values given in the form of a grouped frequency distribution table.

Class Intervals Frequency
0-52
5-103
10-151
15-204
20-255
25-309
30-356
35-4012
40-458
45-507

We calculate the values required to calculate the mean and the standard deviation,

ClassesClass Mark (x_i)Frequency
(f_i)
f_ix_ix_i-\bar{x}f_i(x_i-\bar{x})^2f_i(x_i-\bar{x})^4
0 – 52.525-28.861665.761387376
5-107.5322.5-23.861707.85972249
10-1512.5112.5-18.86355.686126513
15 – 2017.5470-13.86768.36147594
20 – 2522.55112.5-8.8596392.46730806.1
25 – 3027.59247.5-3.8596134.0721997.26
30 – 3532.561951.14047.802410.1462
35 – 4037.5124506.1404452.44717059
40 – 4542.5834011.1404992.859123221
45 – 5047.57332.516.14041823.58475062
n=57\sum f_ix_i=1787.5\sum f_i(x_i-\bar{x})^2 =8300.877 \sum f_i(x_i-\bar{x})^4= 3281887
\text{Mean }\bar{x}=\frac{\sum f_ix_i}{n}=\frac{1787.5}{57} = 31.3596 \text{Standard Deviation }\sigma =\sqrt{\frac{\sum f_i(x_i-\bar{x})^2}{n}}=\sqrt{\frac{8300.8772}{57}} = 12.0677

The formula for calculating kurtosis for a set of grouped data values is as follows,

\text{Kurtosis } = \frac{\sum_{i=1}^{n}f_i(x_i-\bar{x})^4}{n\sigma^4}= \frac{3281887.0786}{57 \times 12.0677^4}= 2.7149

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