Find Sk_{p} for the following set of observations:

18, 27, 10, 25, 31, 13, 28.

#### Solution

The given data can be arranged in ascending order as follows:

10, 13, 18, 25, 27, 28, 31.

Here, n = 7

∴ Median = value of `(("n" + 1)/2)^"th"` observation

= value of `((7 + 1)/2)^"th"` observation

= value of 4^{th} observation

= 25

For finding standard deviation, we construct the following table:

x_{i} |
x_{i}^{2} |

10 | 100 |

13 | 169 |

18 | 324 |

25 | 625 |

27 | 729 |

28 | 784 |

31 | 961 |

152 |
3692 |

From the table, `sum"x"_"i" = 152, sum"x"_"i"^2 = 3692`

Mean = `bar"x" = (sum"x"_"i")/"n" = 152/7` = 21.7143

∴ S.D. = `sqrt((sum"x"_"i"^2)/"n" - (bar"x")^2`

= `sqrt(3692/7 - (21.7143)^2`

= `sqrt(527.4286 - 471.5108)`

= `sqrt(55.9178)`

= 7.4778

Coefficient of skewness,

Sk_{p} = `(3("Mean"-"Median"))/"S.D."`

= `(3(21.7143 - 25))/(7.4778)`

= `(3(-3.2857))/(7.4778)`

= `(-9.8571)/(7.4778)`

∴ Sk_{p} = – 1.3182