Answer:
Kindly check explanation
Explanation:
Given the data:
Number of families __________Expenditure
______midpoint (x) ____Town A __Town B
21-30 __25.5____________3 _______2
31-40 __35.5____________61 ______14
41-50 __45.5___________132 _____ 20
51-60 __55.5__________ 153 ______27
61-70 __65.5__________ 140 ______ 28
71-80 __75.5___________ 51 _______ 7
81-90 __85.5___________ 2 _______ 2
The variability is the variance and standard deviation :
The variance (s):
√Σ(x - m)²/Σf(x)
Town A:
m = Σ(X * f(x)) / Σf ; Σf = 542
Σ x * f(x) = (25.5*3) + (35.5*61) + (45.5*132) + (55.5*153) + (65.5*140) + (75.5*51) + (85.5*2)
= 29931 / 542
= 55. 22
Σf(x - m)² / Σf - 1 = [3(25.5-55.22)^2 + 61(35.5-55.22)^2 + 132(45.5-55.22)^2 + 153(55.5-55.22)^2 + 140(65.5-55.22)^2 + 51(75.5-55.22)^2 + 2(85.5-55.22)^2] / 541
= 76458. 4928 / 541
Variance = 141.32808
Standard deviation = √variance
Standard deviation = √141.32808
Standard deviation = 11.89
TOWN B:
m = Σ(X * f(x)) / Σf ; Σf = 100
Σ x * f(x) = (25.5*2) + (35.5*14) + (45.5*20) + (55.5*27) + (65.5*28) + (75.5*7) + (85.5*2)
= 5490 / 100
= 54.90
Variance :
Σf(x - m)² / Σf - 1 = [2(25.5-54.90)^2 + 14(35.5-54.90)^2 + 20(45.5-54.90)^2 + 27(55.5-54.90)^2 + 28(65.5-54.90)^2 + 7(75.5-54.90)^2 + 2(85.5-54.90)^2] / 99
= 16764 / 99
= 169.33
Standard deviation = √variance
Standard deviation = √169.3333
Standard deviation = 13.013
From the result above, Town A has lesser variability than Town B due to the slightly lower variance and standard deviation values.