Arithmetic Mean

The arithmetic mean is the most frequently used measure of central tendency.

Arithmetic mean of ungrouped data

Example 3.1: The Demand for a Product on each of 20 days was as follows (in units)

3
12
7
17
3
14
9
6
9
12
11
10
1
4
19
7
15
6
12
8

The arithmetic mean of daily demand is:

Finding arithmetic mean of data in a frequency distribution

The arithmetic mean grouped data is given by :

Where n = number of observations and f = frequency

Example 3.2: In our previous example, the frequency distribution would be as shown below:

Daily Demand
x
Frequency
f
fx
1
1
1
3
2
6
4
1
4
6
2
12
7
2
14
8
1
8
9
2
18
10
1
10
11
1
11
12
3
36
14
1
14
15
1
15
17
1
17
19
1
19
Total
20
185

Therefore:

Arithmetic Mean of Grouped Data

Here (below) where x is the class midpoint

Example 3.3:

Using the data on daily demand, the frequency distribution might have been shown as follows:

Daily Demand
Midpoint
X
Frequency
F
fx
0 to less than 5
2.5
4
10.0
5 to less than 10
7.5
7
52.5
10 to less than 15
12.5
6
75.0
15 to less than 20
17.5
3
52.5
Total
-
20
190.0

Therefore:

Here we have the assumption that frequencies occur evenly within each class interval, which is not correct. That is why our approximate mean of 9.5 is not correct and is in error by 0.25. As the frequencies become larger, the size of this approximating error should become smaller.

Weighted Arithmetic Mean

The weighted arithmetic mean can be calculated by using the formula:

^where are the weighting factors.

Example 3.4: Calculate the weighted arithmetic mean for the distribution below.

Weekly Income
(MWK)
Category Average
x
Number of Workers
f
100 - < 200
170
10
200 - < 300
260
28
300 - < 400
350
42
400 - < 600
590
50
600 - < 1000
7500
20
Total
-
150

Solution: