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Worked Examples · Example 7
Q.

Compute Karl Pearson's coefficient of skewness for the following distribution and comment on its shape.

xx510152025
ff371262
Maharashtra MsbshseTextbookSubjectiveImportance★★★★★est
93% · 14/15 Questions
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Setting up the table. With N=∑f=3+7+12+6+2=30N = \sum f = 3 + 7 + 12 + 6 + 2 = 30:

xxfffxfxfx2fx^2
531575
10770700
15121802700
2061202400
252501250
Total304357125

Mean. xˉ=∑fxN=43530=14.5.\bar{x} = \frac{\sum fx}{N} = \frac{435}{30} = 14.5.

Mode. The greatest frequency, 1212, occurs at x=15x = 15, so the mode Z=15Z = 15.

Standard deviation. σ=∑fx2N−xˉ2=712530−(14.5)2=237.5−210.25=27.25≈5.22.\sigma = \sqrt{\frac{\sum fx^2}{N} - \bar{x}^2} = \sqrt{\frac{7125}{30} - (14.5)^2} = \sqrt{237.5 - 210.25} = \sqrt{27.25} \approx 5.22.

Coefficient. Skp=xˉ−Zσ=14.5−155.22=−0.55.22≈−0.096.Sk_p = \frac{\bar{x} - Z}{\sigma} = \frac{14.5 - 15}{5.22} = \frac{-0.5}{5.22} \approx -0.096. The value is negative but very small, so the distribution is only slightly negatively skewed — almost symmetric. …

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