Discrete Standard Deviation
For the calculation of standard deviation in discrete series any of the following techniques may be used:
- Actual mean method.
- Assumed mean method.
- Step deviation method.
1) Actual mean method: When this method is applied then deviations are taken from the actual mean. We find (X-X) and represent these deviations by x. These deviations obtained are then squared and multiplied by the respective frequencies.
The following formula is then applied for calculating standard deviation:
σ = √ Σ fx2 /N where x = (X - X)
However in practice this technique is rarely used because if the actual mean is in fractions the calculations take a lot of time for.
2) Assumed mean method: When this method is used the following formula is applied:
Σ = √ Σfd2 / N - (Σ fd/N)2 where d = (x - A)
Steps:
1) Take the deviations of the items from an assumed mean and represent the deviation by d.
2) Multiply the obtain deviations by the respective frequencies and obtain the total Σ fd.
3) Obtain the squares of the deviations and calculate d2. Multiply the squared deviations by the respective frequencies and obtain the total Σ fd2.
4) Then Substitute the value in the above formula
Illustration:
Calculate the standard deviation from the data given below:
|
Size of item
|
frequency
|
Size of item
|
frequency
|
|
3.5
|
3
|
7.5
|
85
|
|
4.5
|
7
|
8.5
|
32
|
|
5.5
|
22
|
9.5
|
8
|
|
6.5
|
60
|
|
|
Solution:
Calculation of standard deviation
|
X size of item
|
f
|
(X - 6.5) d
|
Fd
|
Fd2
|
|
3.5
|
3
|
-3
|
-9
|
27
|
|
4.5
|
7
|
-2
|
-14
|
28
|
|
5.5
|
22
|
-1
|
-22
|
22
|
|
6.5
|
60
|
0
|
0
|
0
|
|
7.5
|
85
|
+1
|
+85
|
85
|
|
8.5
|
32
|
+2
|
+64
|
128
|
|
9.5
|
8
|
+3
|
+24
|
72
|
|
|
N = 217
|
|
Σ fd = + 128
|
Σ fd2 = 362
|
Σ √Σfd2/N - (Σ fd / N)2
Σ fd2 = 362, Σ fd = 128 N= 217
Σ = √ 362 / 217 - (128 / 127)2 = √ 1.668 - 348 = 1.149
3) Step deviation method: In this method we take deviations of midpoints from an assumed mean and divide these deviations by the width of class interval. If class intervals are unequal we divide the deviations of midpoints by the lowest common factor and use x instead of I in the formula for the calculation of standard deviation. The formula for calculating standard deviation is as follows:
Σ fd2 = 362 Σ fd /N)2 x I
Where d = (X -A) /I and I = class interval.
The use of the above formula simplifies calculations.