Examples of Arithmetic Mean

Example:

The following data shows distance covered by 100 people to perform their routine jobs.

Distance (Km)
0 - 10
10 - 20
20 - 30
30 - 40
Number of People
10
20
40
30

Calculate the arithmetic mean by step-deviation method; also explain why it is better than direct method in this particular case.

 

Solution:

The given distribution is grouped data and the variable involved is distance covered, while the number of people represents frequencies.

Distance Covered in (Km)
Number of People
f
Mid Points
x
u = \left( {\frac{{x - 5}}{{10}}} \right)
fu
0 - 10
10
5
 - 1
 - 10
10 - 20
20
15
0
0
20 - 30
40
25
 + 1
40
30 - 40
30
35
 + 2
60
Total
\sum f = 100
\sum fu = 90

Now we will find the arithmetic mean as \overline X = A + \frac{{\sum fu}}{{\sum f}} \times h
Here
A = 15,    \sum fu = 90,    \sum f = 100   and   h = 10
\overline X = 15 + \frac{{90}}{{100}} \times 10 = 24 Km

Explanation:

Here from the mid points (x) it is very clear that each mid point is a multiple of 5 and there is also a gap of 10 from mid point to midpoint, i.e. class size or interval (h). Keeping this in mind, we should use the step-deviation method instead of direct method.

 

Example:

The following frequency distribution showing the marks obtained by 50 students in statistics at a certain college. Find the arithmetic mean using (1) direct method (2) short-cut method (3) step-deviation.

Marks
20 - 29
30 - 39
40 - 49
50 - 59
60 - 69
70 - 79
80 - 89
Frequency
1
5
12
15
9
6
2

 

Solution:

Direct Method
Short-Cut
Method
Step-Deviation
Method
Marks
f
x
fx
D = x - A
fD
u = \frac{{x - A}}{h}
fu
20 - 29
1
24.5
24.5
 - 30
 - 30
 - 3
 - 3
30 - 39
5
34.5
172.5
 - 20
 - 100
 - 2
 - 10
40 - 49
12
44.5
534.5
 - 10
 - 120
 - 1
 - 12
50 - 59
15
54.5
817.5
0
0
0
0
60 - 69
9
64.5
580.5
10
90
1
9
70 - 79
6
74.5
447.5
20
120
2
12
80 - 89
2
84.5
169.5
30
60
3
6
Total
50
2745
20
2

 

(1) Direct Method:

\overline X = \frac{{\sum fx}}{{\sum f}} = \frac{{2745}}{{50}} = 54.9 or 55 Marks

(2) Short-Cut Method:
\overline X = A + \frac{{\sum fD}}{{\sum f}}       Where A = 54.5
 = 54.5 + \frac{{20}}{{50}} = 54.5 + 0.4 = 54.9Marks

(3) Step-Deviation Method:
\overline X = A + \frac{{\sum fu}}{{\sum f}} \times h   Where A = 54.5        h = 10

 = 54.5 + \frac{2}{{50}} \times 10

 = 54.5 + 0.4 = 54.9 Marks