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Calculate Skb for the following set of observations of yield of wheat in kg from 13 plots:4.6, 3.5, 4.8, 5.1, 4.7, 5.5, 4.7, 3.6, 4.2, 3.5, 3.6, 5.2 - Mathematics and Statistics

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Sum

Calculate Skb for the following set of observations of yield of wheat in kg from 13 plots:
4.6, 3.5, 4.8, 5.1, 4.7, 5.5, 4.7, 3.6, 4.2, 3.5, 3.6, 5.2

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Solution

The given data can be arranged in ascending order as follows:
3.5, 3.5, 3.5, 3.6, 3.6, 4.2, 4.6, 4.7, 4.7, 4.8, 5.1, 5.2, 5.5
Here, n = 13

Q1 = value of `(("n" + 1)/4)^"th"` observation

= value of `((13  + 1)/4)^"th"` observation
= value of (3.50)th observation
= value pf 3rd observation + 0.5 (value of 4th observation – value of 3rd observation)
= 3.5 + 0.50 (3.6 – 3.5)
= 3.5 + 0.50 × 0.1
= 3.5 + 0.05
∴ Q1  = 3.55

Q2 = value of 2`(("n" + 1)/4)^"th"` observation

= value of 2`((13 + 1)/4)^"th"` observation

= value of (2 × 3.50)th observation

= value of 7th observation
∴ Q2 = 4.6

Q3 = value of 3`(("n" + 1)/4)^"th"` observation

= value of `3((13+1)/4)^"th"` observation
 = value of (3 × 3.50)th observation
= value of (10.50)th observation
= value of 10th observation + 0.5 (value of 11th observation – value of 10th observation)
= 4.8 + 0.50 (5.1 – 4.8)
= 4.8 + 0.50 × 0.3
= 4.8 + 0.15
∴ Q3  = 4.95

∴ Skb = `("Q"_3 + "Q"_1 - 2"Q"_2)/("Q"_3 - "Q"_1)`

= `(4.95 + 3.55 - 2(4.6))/(4.95 + 3.55)`

= `(8.5 - 9.2)/(1.4)`

= `(-0.7)/(1.4)`
∴ Skb= – 0.5

Concept: Measures of Skewness - Bowley’s Coefficient of Skewness
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