Statistics XI


   
 
Standard Deviation
Standard Deviation of a statistical data is defined as the positive square root of the arithmetic mean of the squared deviations of items from their arithmetic mean of the series under consideration.
 
The standard deviation is denoted by s (sigma).
 
The standard deviation is also known as “Root Mean Square Deviation” because it is the square root of the arithmetic mean of the squares of the deviation.
 
Square of the standard deviation is called the variance.
 
For an individual series,
 
 
For a frequency distribution,
 
 
 
The square of the Standard deviation is known as Variance.
 
 
Variance = (S.D.)2
 
 
Let  s1 and s2 be the S.D of the two groups containing n1 and n2 items respectively. Let and be their respective A.M. Let x and s be the A.M and S.D of the combined group respectively.
 
 
Methods of calculation of standard deviation (S.D = s)
 
Individual series
 
There are two methods
 
Deviations are taken from actual mean
 
Deviations are taken from assumed mean
 
Deviations taken from actual mean
 
Step 1:
 
Calculate the mean.
 
Step 2 :
 
Take deviations of the item from the calculated mean and call it x.
 
Step 3:
 
Find the squares of these deviations i.e., x2 and obtain the sum of the square of deviation i.e.
 
Step 4:
 
Divide the total obtained in step 3 by the number of items and calculate the square root to get the standard deviation.
 
Example:
 
Calculate the standard deviation and the variance for the following data
 
7, 8, 11, 6, 13, 8, 10.
 
Suggested answer:
 
 
Arrange the data in the increasing order or decreasing order (for convenience).
 
Deviation from actual mean:
 
 
 
Deviation from assumed mean:
 
 
 
 
Deviations taken from assumed mean
 
Step 1:
 
Take the deviations of each item from the assumed mean and call it d and find S d.
 
Step 2:
 
Calculate the squares of these deviations and then apply the formula
 
 
where
 
 
  
     
   
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