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Step-deviation Method

When the values of the mid-values xi in the different classes of a grouped data are large in magnitude, then the computation of the mean x becomes quite lengthy and tedious. In such a case, computation is simplified by 'short-cut method'. The procedure is,
 

1. Choose an arbitrary constant 'a' (also called assumed mean).
2. Subtract 'a' from the values of xi.
3. The reduced value (xi - a) is called the deviation of xi from 'a'.

4. Divide the deviations (xi - a) by a constant h (h is taken to be width of the class interval in the frequency-table (or) h = difference in two successive value of xi).
5. 'a' is taken some where in the middle of all values of xi.

Now, we can define
ui=

h = difference in two successive values of xi 
Now we know that,


Mean=
We can also calculate the mean by the formula.
x =
where 'a' is assumed mean and di represents the deviation from the assumed mean i.e.
di = xi - a.

 

Example

For the following distribution, find out the mean-wage.

Wages in
Rupees
900 950 1000 1100 1260 1440 1500
No. of
workers
26 22 18 19 15 3 2
 
Solution

Let the assumed mean 'a' = 1100
Now we may arrange the given data as under:

Wages (in Rs) Frequency fi Deviation
di=x- a =x- 1100
fi di
900 26 -200 -5200
950 22 -150 -3300
1000 18 -100 -1800
1100 = a 19 00 00
1260 15 160 2400
1440 3 340 1020
1500 2 400 800
Total Σfi = 105   Σfix= -6080

Mean= = 1100 -  = 1042.1.


 

Example

The frequency distribution of marks in mathematics are given in the table. Find the mean by short-cut method.

Marks 5 10 15 20 25 30 35 40 45 50
No. of 
Students
15 50 80 76 72 45 39 9 8 6
Solution

Let the assumed mean = a = 25
h = 10 - 5 = 15 - 10 = … = 50 - 45 = 5
Now we have the following table,

Marks xi No. of students fi x- a ui
Here h=5
fiui
5 15 -20 -4 -60
10 50 -15 -3 -150
15 80 -10 -2 -160
20 76 -5 -1 -76
25 = a 72 0 0 0
30 45 5 1 45
35 39 10 2 78
40 9 15 3 27
45 8 20 4 32
50 6 25 5 30
Total N = 400     -234


Now 

         = 

         =- 0.585   
 

= 25 + (5 × −0.585)
                   = 25 - 2.925 = 22.075

Mean marks = 22.075.

 

 

Example

 In a study on a certain disease, the following data was given. Find the average age of the first  detection.

Age of first detection (in yrs) No. of patients
4 - 8 2
8 - 12 12
12 - 16 15
16 - 20 25
20 - 24 18
24 - 28 12
28 - 32 3
32 - 36 1
Solution

Let the assumed mean a = 18 and h = 4.
Then we have the following table for the solution.

 

Age in 
Years
Mid-point
xi
Frequency
fi
x- a ui= 
Here h = 4
fiui
4 - 8 6 2 -12 -3 -6
8 - 12 10 12 -8 -2 -24
12 - 16 14 15 -4 -1 -15
16 - 20 18 = a 25 0 0 0
20 - 24 22 18 4 1 18
24 - 28 26 12 8 2 24
28 - 32 30 3 12 3 9
32 - 36 34 1 16 4 4
Total   Σfi = 88     Σfiu= 10


      (where N = Σfi)

u =  = 0.114
 = 18 + (4 
× 0.114)
Mean      =18 + 0.46 = 18.46.


 

Example

Find the mean of the following data:

x 4 7 10 13 16 19 22
f 23 25 27 29 27 25 23
Solution

Let the assumed mean = a = 13, then we have the following table:

xi fi d= x- a di fi
4 23 -9 -207
7 25 -6 -150
10 27 -3 -81
13 29 00 00
16 27 +3 +81
19 25 +6 +150
22 23 +9 +207
  Σfi = 179   Σfid= 0

Mean =  = 13 +  =13

 

Second Method of Solution
Hint: a = 13, h = 3, ui


Here Σfi = 179 and Σfiui = 0
= 13 + (3 × ) = 13.







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