Statistics - Kurtosis


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The degree of flatness or peakedness is measured by kurtosis. It tells us about the extent to which the distribution is flat or peak vis-a-vis the normal curve. Diagrammatically, shows the shape of three different types of curves.

kurtosis

The normal curve is called Mesokurtic curve. If the curve of a distribution is more peaked than a normal or mesokurtic curve then it is referred to as a Leptokurtic curve. If a curve is less peaked than a normal curve, it is called as a platykurtic curve. Kurtosis is measured by moments and is given by the following formula:

Formula

${\beta_2 = \frac{\mu_4}{\mu_2}}$

Where −

  • ${\mu_4 = \frac{\sum(x- \bar x)^4}{N}}$

The greater the value of \beta_2 the more peaked or leptokurtic the curve. A normal curve has a value of 3, a leptokurtic has \beta_2 greater than 3 and platykurtic has \beta_2 less then 3.

Example

Problem Statement:

The data on daily wages of 45 workers of a factory are given. Compute \beta_1 and \beta_2 using moment about the mean. Comment on the results.

Wages(Rs.)Number of Workers
100-2001
120-2002
140-2006
160-20020
180-20011
200-2003
220-2002

Solution:

Wages
(Rs.)
Number of Workers
(f)
Mid-pt
m
m-${\frac{170}{20}}$
d
${fd}$${fd^2}$${fd^3}$${fd^4}$
100-2001110-3-39-2781
120-2002130-2-48-1632
140-2006150-1-66-66
160-2002017000000
180-20011190111111111
200-200321026122448
220-2002230361854162
 ${N=45}$  ${\sum fd = 10}$${\sum fd^2 = 64}$${\sum fd^3 = 40}$${\sum fd^4 = 330}$

Since the deviations have been taken from an assumed mean, hence we first calculate moments about arbitrary origin and then moments about mean. Moments about arbitrary origin '170'

${\mu_1^1= \frac{\sum fd}{N} \times i = \frac{10}{45} \times 20 = 4.44 \\[7pt] \mu_2^1= \frac{\sum fd^2}{N} \times i^2 = \frac{64}{45} \times 20^2 =568.88 \\[7pt] \mu_3^1= \frac{\sum fd^2}{N} \times i^3 = \frac{40}{45} \times 20^3 =7111.11 \\[7pt] \mu_4^1= \frac{\sum fd^4}{N} \times i^4 = \frac{330}{45} \times 20^4 =1173333.33 }$

Moments about mean

${\mu_2 = \mu'_2 - (\mu'_1 )^2 = 568.88-(4.44)^2 = 549.16 \\[7pt] \mu_3 = \mu'_3 - 3(\mu'_1)(\mu'_2) + 2(\mu'_1)^3 \\[7pt] \, = 7111.11 - (4.44) (568.88)+ 2(4.44)^3 \\[7pt] \, = 7111.11 - 7577.48+175.05 = - 291.32 \\[7pt] \\[7pt] \mu_4= \mu'_4 - 4(\mu'_1)(\mu'_3) + 6 (\mu_1 )^2 (\mu'_2) -3(\mu'_1)^4 \\[7pt] \, = 1173333.33 - 4 (4.44)(7111.11)+6(4.44)^2 (568.88) - 3(4.44)^4 \\[7pt] \, = 1173333.33 - 126293.31+67288.03-1165.87 \\[7pt] \, = 1113162.18 }$

From the value of movement about mean, we can now calculate ${\beta_1}$ and ${\beta_2}$:

${\beta_1 = \mu^2_3 = \frac{(-291.32)^2}{(549.16)^3} = 0.00051 \\[7pt] \beta_2 = \frac{\mu_4}{(\mu_2)^2} = \frac{1113162.18}{(546.16)^2} = 3.69 }$

From the above calculations, it can be concluded that ${\beta_1}$, which measures skewness is almost zero, thereby indicating that the distribution is almost symmetrical. ${\beta_2}$ Which measures kurtosis, has a value greater than 3, thus implying that the distribution is leptokurtic.



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