]]>
LearnNext
Get a free home demo of LearnNext

Available for CBSE, ICSE and State Board syllabus.
Call our LearnNext Expert on 1800 419 1234 (tollfree)
OR submit details below for a call back

clear

Shortcut Method To Find Variance and Standard Deviation

2,584 Views
Have a doubt? Clear it now.
live_help Have a doubt, Ask our Expert Ask Now
format_list_bulleted Take this Lesson Test Start Test

Shortcut Method To Find Variance and Standard Deviation - Lesson Summary

Variance of a discrete frequency distribution:   Observation (xi)   x 1   x 2   x 3   ...   x n   Frequency (fi)   f 1   f 2   f 3   ...   f n
σ 2 = ∑ i = 1 n f i x i - x _ 2 N

Total of the frequencies: N = ∑ i = 1 n f i
 
Variance of a continuous frequency distribution:



σ 2 = ∑ i = 1 n f i ( x i - x _ ) 2 N

Total of the frequencies: N = ∑ i = 1 n f i
 
Many a times, the values of x i in the discrete frequency distribution or the mid-values in the continuous distribution are large. In such cases, there are heavy calculations to find the mean, variance, standard deviation, and so on. To avoid such tedious and time-consuming calculations, step-deviation method by working with an assumed mean is used.
 
Let the assumed mean be A.
The deviations of the mid-values can be reduced by 1 h  times, where h is the common factor among the mid-values (we usually take the width of the class intervals).
Let the step-deviations be y i =  x i - A h

⇒  x i = A + h y i                     .... eq (1)

x _ = ∑ i = 1 n f i x i N

x _   = ∑ i = 1 n f i (A + h y i ) N

  = A ∑ i = 1 n f i N + h ∑ i = 1 n f i y i N

  =  AN N + h ∑ i = 1 n x i y i N                             (∵ ∑ i = 1 n f i   = N  )

Let   ∑ i = 1 n f i N   =  y _

∴  x _ = A + h y _                          .... eq(2)
Substituting eq (1) and eq (2) in the formula for the variance, σ 2 = ∑ i = 1 n f i ( x i - x _ ) 2 N ,

σ 2 = ∑ i = 1 n f i ( A + hy i - A -  hy _ ) 2 N

σ 2 = ∑ i = 1 n f i (  hy i -  h  y _ ) 2 N

σ 2 = h 2 N ∑ i = 1 n f i ( y i - y _ ) 2

The variance deduced is in the term of variable y i.
 
Hence, this can also be written as: σ x 2 = h 2σ y 2.
 
⇒ σ x = hσ y

σ x  = N ∑ f i y i 2 - ( ∑ i = 1 n f i y i ) 2

  (∵ σ y  =  1 N N ∑ f i y i 2 - ( ∑ i = 1 n f i y i ) 2 )

Comments(0)

Feel the LearnNext Experience on App

Download app, watch sample animated video lessons and get a free trial.

Desktop Download Now
Tablet
Mobile
Try LearnNext at home

Get a free home demo. Book an appointment now!

GET DEMO AT HOME