1. Arithmetic Means

2. Introduction to Central Tendency
Statistical diagrams help us to organize and present a set of data.
However, we can also represent a set of data by just one number,
which tells us the central tendency of the data set. Such kind of numbers are called averages.
Three commonly-used averages are arithmetic mean, the median and the mode.

3. Arithmetic Means of Ungrouped Data
The arithmetic mean (or mean) can be calculated as follows:
the arithmetic mean of 4, 8, 1 and 3
For example, 4 3 1 8 + = 4 16 = 4 =

4. Follow-up question
Find the arithmetic means for the following sets of data. (a) 22, 28, 31, 47 (b) –1, 0, 3, 5, 8, 10, 10
Solution

5. If the data are presented in a frequency distribution table, we can find the arithmetic mean as follows:
where x is a particular datum and f is the frequency of the datum.

6. For example,
the following table shows the number of story books read by some students in a week.
No. of story books x 1 2 3
No. of students f 12 7 6
read
books
story of
number
Mean 6 7 12 3 2 1 +  = 25 44 =
1.76 =

7. Follow-up question
The following table shows the number of toys sold by some shops on a day.
No. of toys 5 6 7 8
No. of shops 3 4
Find the mean number of toys sold by the shops on that day.
Solution
sold
toys of
number
Mean 4 6 3 7 8 5 +  = 20
127 =
6.35 =

8. Arithmetic Means of Grouped Data
If there are a large number of continuous data covering a wide range, we usually organize the data into groups.
For example,
the following table shows the weights (in kg) of some students.
We can construct a frequency distribution table to present the above data.

9.
Weights (kg)
Class mark x (kg)
Frequency f
xf (kg)
40 – 49
50 – 59
60 – 69
70 – 79
Total
44.5 4
178
54.5 6
327
64.5 6
387
74.5 4
298 20
1190 20
1190
Mean weight of the students = kg
59.5 = kg

10. the arithmetic mean obtained is an approximation only.
Weights (kg)
Class mark x (kg)
Frequency f
xf (kg)
40 – 49
50 – 59
60 – 69
70 – 79
Total
44.5
44.5 4
178
54.5
54.5 6
327
64.5
64.5 6
387
74.5
74.5 4
298 20
1190 20
1190
Mean weight of the students = kg
59.5 = kg
When we use the class mark to represent the data within a class interval,
the arithmetic mean obtained is an approximation only.

11. Follow-up question
The following table shows the age distribution of 20 staff in a company.
Age
Class mark x
Frequency f xf
20 – 29
30 – 39 12
40 – 49 3
Total
24.5
20 – 12 – 3
= 5
122.5
34.5
414
44.5
133.5 20
670
Complete the table and, hence find the mean age of the staff.
Solution 20
670
staff
the of
age
Mean =
33.5 =