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If a curve is relatively more narrow and peaked at

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Unformatted text preview: frequency curve is termed as kurtosis. Measure of kurtosis denote the shape of top of a frequency curve. Measure of kurtosis tell us the extent to which a distribution is more peaked or more flat topped than the normal curve, which is symmetrical and bell-shaped, is designated as Mesokurtic. If a curve is relatively more narrow and peaked at the top, it is designated as Leptokurtic. If the frequency curve is more flat than normal curve, it is designated as platykurtic. 184 L = Lepto Kurtic M = Meso Kurtic P = Platy Kurtic 7.11.1 Measure of Kurtosis: The measure of kurtosis of a frequency distribution based moments is denoted by β2 and is given by If β2 =3, the distribution is said to be normal and the curve is mesokurtic. If β2 >3, the distribution is said to be more peaked and the curve is leptokurtic. If β2< 3, the distribution is said to be flat topped and the curve is platykurtic. Example 24: Calculate β1 and β2 for the following data. X: 0 12 3 4 5 6 7 8 F: 5 10 15 20 25 20 15 10 5 Solution: [Hint: Refer Example of page 172 and get the values of first four central moments and then proceed to find β1 and β2] ∑ fd 2 500 = 0 = = =4 µ1 µ2 N 125 ∑ fd 3 ∑ fd 4 4700 =0 = = = 37.6 = µ4 µ3 N N 125 0 =0 64 185 2 = 4 2 2 = 37.6 42 37.6 = 2.35 16 The value of β2 is less than 3, hence the curve is platykurtic. = Example 25: From the data given below, calculate the first four moments about an arbitrary origin and then calculate the first four central moments. X : 30-33 33-36 36-39 39-42 42-45 45-48 f: 2 4 26 47 15 6 Solution: [Hint: Refer Example 18 of page 172 and get the values of first four moments about the origin and the first four moments about the mean. Then using these values find the values of β1 and β2.] µ1 = 0, µ2 = 8.76 µ3 = −2.91, µ4 = 291.454 β1 = 8.47 (−2.91) 2 = 0.0126 = 3 672.24 (8.76) 291.454 = 3.70 (8.76) 2 Since β2 >3, the curve is leptokurtic. β2 = Exercise – 7 I. Choose the best answer: 1. Which of the following is a unitless measure of dispersion? (a) Standard devi...
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