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Standard Deviation

Standard Deviation is the positive square root of the mean of squared deviations from mean. So if there are five values x1, x2, x3, x4 and x5, first their mean is calculated. Then deviations of the values from mean are calculated. These deviations are then squared. The mean of these squared deviations is the variance. Positive square root of the variance is the standard deviation. (Note that Standard Deviation is calculated on the basis of the mean only).

Calculation of Standard Deviation for ungrouped data:
Four alternative methods are available for the calculation of standard deviation of individual values. All these methods result in the same value of standard deviation. These are:
  1. Actual Mean Method
  2. Assumed Mean Method
  3. Direct Method
  4. Step-Deviation Method

Actual Mean Method

Suppose you have to calculate the standard deviation of the following values:
5, 10, 25, 30, 50

Example 8
Following formula is used:
s = = = 15.937

Do you notice the value from which deviations have been calculated in the above example? Is it the Actual Mean?
 

X

d

d2

5

-19

361

10

-14

196

25

1

1

30

6

36

50

26

676

   

1270

Assumed Mean Method

For the same values, deviations may be calculated from any arbitrary value

Example 9

 

X

d

d2

5

-20

400

10

-15

225

25

0

0

30

5

25

50

25

625

   

1275


Formula for Standard Deviation

                            = 15.937

The sum of deviations from a value other than actul mean is not equal to zero

Direct Method

Standard Deviation can also be calculated from the values directly,
i.e., without taking deviations, as shown below:

Example 10
 

X

X2

5

25

10

100

25

625

30

900

50

2500

 

4150

 

(This amounts to taking deviations from zero)
Following formula is used.

= 15,937

Standard Deviation is not affected by the value of the constant from which deviations are calculated. The value of the constant does not figure in the standard deviation formula. Thus, Standard Deviation is Independent of Origin.

Step-deviation Method

If the values are divisible by a common factor, they can be so divided and standard deviation can be calculated from the resultant values as follows:

Example 11
Since all the five values are divisible by a common factor 5, we divide and get the following values:


(Steps in the calculation are same as in actual mean method).

The following formula is used to calculate standard deviation:

c = common factor
Substituting the values,

= 15.937

Alternatively, instead of dividing the values by a common factor, the deviations can be divided by a common factor. Standard Deviation can be calculated as shown below:

Example 12
 


Deviations have been calculated from an arbitrary value 25. Common factor of 5 has been used to divide deviations.

                       = 15.937
Standard Deviation is not independent of scale. Thus, if the values or deviations are divided by a common factor, the value of the common factor is used in the formula to get the value of Standard Deviation.
Standard Deviation in Continuous frequency distribution: Like ungrouped data, S.D. can be calculated for grouped data by any of the following methods:
  1. Actual Mean Method
  2. Assumed Mean Method
  3. Step-Deviation Method

Actual Mean Method


For the values in Table 6.2, Standard Deviation can be calculated as follows:

Example 13


Following steps are required:
  1. Calculate the mean of the distribution.
    = 40.5
  2. Calculate deviations of mid-values from the mean so that
    (Col. 5)
  3. Multiply the deviations with their corresponding frequencies to get
    'fd' values (col. 6) [Note that Σ fd
    = 0]
  4. Calculate '' values by multiplying 'fd' values with 'd' values. (Col. 7). Sum up these to get .
  5. Apply the formula as under:

        = 17.168

Assumed Mean Method


For the values in example 13, standard deviation can be calculated by taking deviations from an assumed mean (say 40) as follows:

Example 14


The following steps are required:
  1. Calculate mid-points of classes
  2. Calculate deviations of mid-points from an assumed mean such that
    d = m - (Col. 4). Assumed Mean = 40.
  3. Multiply values of 'd' with corresponding frequencies to get 'fd' values (Col. 5). (note that the total of this column is not zero since deviations have been taken from assumed mean).
  4. Multiply 'fd' values (Col. 5) with'd' values (col. 4) to get fd2 values (col. 6). Find.
  5. Standard Deviation can be calculated by the following formula.

                           = 17.168

Step-deviation Method


In case the values of deviations are divisible by a common factor, the calculations can be simplified by the step-deviation method as in the following example.

Example 15


Steps required:
  1. Calculate class mid-points (Col. 3) and deviations from an arbitrarily chosen value, just like in the assumed mean method. In this example, deviations have been taken from the value 40. (Col. 4)
  2. Divide the deviations by a common factor denoted as 'C'. C = 5 in the above example. The values so obtained are 'd'' values (Col. 5).
  3. Multiply 'd'' values with corresponding 'f'' values (Col. 2) to obtain 'fd'' values (Col. 6).
  4. Multiply 'fd'' values with 'd'' values to get 'fd'2' values (Col. 7)
  5. Sum up values in Col. 6 and Col.
    7 to get Σ fd' and Σ fd'2 values.
  6. Apply the following formula.
Standard Deviation: Comments
Standard Deviation, the most widely used measure of dispersion, is based on all values. Therefore a change in even one value affects the value of standard deviation. It is independent of origin but not of scale. It is also useful in certain advanced statistical problems.




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