Question

Obtain the coefficient of skewness by formula and comment on nature of the distribution.

Height in inches	No. of females
Less than 60	10
60 – 64	20
64 – 68	40
68 – 72	10
72 – 76	2

Answer 1

We construct the less than cumulative frequency table as given below.

Height in inches	No. of females (f)	Less than cumulative frequency (c.f.)
Less than 60	10	10
60 – 64	20	30 ← Q1
64 – 68	40	70 ← Q2, Q3
68 – 72	10	80
72 – 76	2	82
Total	N = 82

Q₁ class = class containing `(("N")/4)^"th"` observation

∴ `"N"/(4) = (82)/(4)` = 20.5

Cumulative frequency which is just greater than (or equal) to 20.5 is 30.
∴ Q₁ lies in the class 60 – 64.
L = 60, f = 20, c.f. = 10, h = 4

∴ Q₁ = `"L"+"h"/"f"("N"/4-"c.f.")`

= `60 + (4)/(20)(20.5 - 10)`

= `60 + (1)/(5) xx 10.5`

= 60 + 2.1
∴ Q₁ = 62.1

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

∴ `"N"/2 = (82)/(4)` = 41

Cumulative frequency which is just greater than (or equal) to 41 is 70.

∴ Q₂ lies in the class 64 - 68

∴ L = 64, h = 4, f = 40, c.f. = 30

∴ Q₂ = `"L"+"h"/"f"("N"/2-"c.f.")`

= `64+4/40(41-30)`

= `64+1/10(11)`

 = 64 + 1.1
∴ Q₂ = 65.1

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

∴ `(3"N")/(4)=(3xx82)/4` = 61.5

Cumulative frequency which is just greater than (or equal) to 61.5 is 70.
∴ Q₃ lies in the class 64 – 68
L = 64, f = 40, c.f. = 30, h = 4

∴ Q₃ = `"L"+"h"/"f"((3"N")/4-"c.f.")`

= `64 + (4)/(40)(61.5 - 30)`

= `64 + (1)/(10) xx 31.5`

= 64 + 3.15
∴ Q₃ = 67.15

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

= `(67.15 + 62.1 - 2(65.1))/(67.15 - 62.1)`

= `(129.25 - 130.2)/(5.05)`

= `(-0.95)/(5.05)`
∴ Sk_b = – 0.1881
Since, Sk_b < 0, the distribution is negatively skewed.