Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 5.57

Solution:

Table : Calculation of Standard deviation

Weight (kg.) mid. pt. Frequency devn.

x f (d x 49= − ) d′=d 2 fd′ fd′^2

44–46 45 3 –4 –2 –6 12

46–48 47 24 –2 –1 –24 24

48–50 49 27 0 0 0 0

50–52 51 21 2 1 21 21

52–54 53 5 4 2 10 20

Total – 80 – – 1 77

Let A (assumed mean) = 49

fd^2 fd^2

i

f f

 ′  ′ 

σ =  −  ×

   

∑ ∑

∑ ∑^

77 1 2

= 80 −   80 × 2

 

0.9625^12

6400

=  − ×

 

= 0.9625 - 0.00016 × 2= 0.96234 2× = 0.9809 × 2 = 1.96 kg.

5.3.2.3.2. MATHEMATICAL PROPERTIES OF STANDARD DEVIATION :
Combined Standard Deviation.
We can also calculate the combined standard deviation for two or more groups, similar to mean of
composite group. The required formula is as follows :

1 1^2 2 2^2 1 1^2 2 2^2

(^1212)
n n n d n d
n n
σ = σ + σ + +

• where σ 12 = combined standard deviation of two groups.
  σ 1 = standard deviation of 1st group.
  σ 2 = standard deviation of 2nd group.
  d 1 =x 1 −x ; d 12 2 =x 2 −x , 12 where
  12 1 1 2 2
  1 2
  x n x n x
  n n
  = +


• 
  

  For Three Groups
  1 1^2 2 2^2 3 3^2 1 1^2 2 2^2 3 3^2
  123
  1 2 3
  n n n n d n d n d
  n n n
  σ = σ + σ + σ + + +

  
  


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