Calculate Bowley’s coefficient of skewness Skb from the following data: Marks above 0 10 20 30 40 50 60 70 80 No. of students 120 115 108 98 85 60 18 5 0 - Mathematics and Statistics

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Sum

Calculate Bowley’s coefficient of skewness Skb from the following data:

Marks above 0 10 20 30 40 50 60 70 80
No. of students 120 115 108 98 85 60 18 5 0
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Solution

To calculate Bowley’s coefficient of skewness Skb, we construct the following table:

Marks above No. of students
‘more than’
(c.f.)
Marks Frequency (fi) Less than cumulative frequency (c.f.)
0 120 0 – 10 5 5
10 115 10 – 20 7 12
20 108 20 – 30 10 22
30 98 30 – 40 13 35 ← Q1
40 85 40 – 50 25 60 ← Q2
50 60 50 – 60 42 102 ← Q3
60 18 60 – 70 13 115
70 5 70 – 80 5 120
80 0 80 – 90 0 120
    Total 120

Here, N = 120
Q1 class = class containing the `("N"/4)^"th"` observation

∴ `"N"/4 = 120/4` = 30
Cumulative frequency which is just greater than (or equal to) 30 is 35.
∴ Q1 lies in the class 30 – 40.
∴ L = 30, h = 10, f = 13, c.f. = 22

∴ Q1 = `"L" + "h"/"f"("N"/4 - "c.f.")`

= `30 + 10/13(30 - 22)`

= `30 + 10/13(8)`

= 30 + 6.1538
∴ Q1 = 36.1538

Q2 class = class containing the `("N"/2)^"th"` observation

∴ `"N"/2 = 120/2` = 60
Cumulative frequency which is just greater than (or equal to) 60 is 60.
∴ Q2 lies in the class 40 – 50.
∴ L = 40, h = 10, f = 25, c.f. = 35

∴ Q2 = `"L" + "h"/"f"("N"/2 - "c.f.")`

= `40 + 10/25 (60 - 35)`

= `40 + 10/25(25)`

∴ Q= 50

Q3 class = class containing the `((3"N")/4)^"th"` observation

∴ `(3"N")/4 = (3 xx 120)/4` = 90
Cumulative frequency which is just greater than (or equal to) 90 is 102.
∴ Q3 lies in the class 50 – 60.
∴ L = 50, h = 10, f = 42, c.f. = 60

∴ Q3 = `"L" + "h"/"f"((3"N")/4 - "c.f.")`

= `50 + 10/42(90 - 60)`

= `50 + 10/42(30)`

= 50 + 7.1429
∴ Q3 = 57.1429
Bowley’s coefficient of skewness:

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

= `(57.1429 + 36.1538 - 2(50))/(57.1429 - 36.1538)`

= `(93.2967 - 100)/(20.9891)`

= `(-6.7033)/(20.9891)`

∴ Skb = – 0.3194

Concept: Measures of Skewness - Bowley’s Coefficient of Skewness
  Is there an error in this question or solution?
Chapter 3: Skewness - Miscellaneous Exercise 3 [Page 44]

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