1. Comparing Different Types of Average

2. Choosing the most Appropriate Measure of Central Tendency
I have learnt three kinds of averages to reflect central tendency. May I use any one of them for any situations?
No, each average has its characteristics. Let’s see how to choose the most appropriate averages for different cases.

3. Arithmetic mean Arithmetic mean is the most commonly used average.
However, if there are extreme values, it may not be appropriate to use arithmetic mean to reflect the central tendency of the data.
Arithmetic mean is the most commonly used average.
All the data are involved in the calculation.
For example,
the mean of the data set {0, 1, 2, 2, 3, 4} = 2.
If the datum ‘4’ is changed to ‘40’ (i.e. {0, 1, 2, 2, 3, 40}),
then the mean = 8,
which is larger than the other 5 data,
so the mean is not a good average to reflect the central
tendency of the data in this case.

4. Median
However, it may be quite time-consuming to arrange a large amount of data in ascending order.
Median is a measure of central tendency which is not affected by extreme values.
For example,
median
for the data set {0, 1, 2, 2, 3, 4} in the previous page,
the median = 2.
median
After changing the datum ‘4’ to ’40’ (i.e. {0, 1, 2, 2, 3, 40}),
the median = 2,
which is not affected by the extreme value,
so it is a usually good average to reflect the central tendency of the
data in this case.

5. Mode
However, it may not lie close to the middle of the distribution. Sometimes, the mode may not exist, or it may not be unique.
Mode or model class represent data which occur most often and they are not affected by extreme values.
For example,
for the data set {0, 0, 2, 3, 4, 5},
the mode = 0,
which cannot show the central tendency of the data.
So the mode is not a good average in this case.

6. The following table summarizes the characteristics of the three averages:
Arithmetic mean
Median
Mode
1. All the data are involved in the calculation
2. Affected by data which are extremely large or extremely small
3. Must have on value only
4. Data have to be arranged in order before calculation
5. Must be one of the data
1. All the data are involved in the calculation   
2. Affected by data which are extremely large or extremely small   
3. Must have one value only   
4. Data have to be arranged in order before calculation   
5. Must be one of the data   

7. Here are some daily-life applications for the three averages in reflecting central tendency.
Arithmetic mean
School test results
Monthly temperature of a city
Median
Salaries of workers
Admission grades of university students
Mode
Best-selling item in a shop
Voting results

8. Follow-up question
The maximum speeds (in km/h) of 5 cars are shown below:
80, 80, 95, 100, 210
(a) Find the arithmetic mean, the median and the mode of the data.
(b) Which of the averages best reflects the central tendency of the maximum speeds of the cars? Explain your answer briefly.
Solution 5
210
100 95 80 + =
(a) Arithmetic mean
km/h
km/h
113 =
Median
km/h 95 =
Mode
km/h 80 =

9. Follow-up question (cont’d)
The maximum speeds (in km/h) of 5 cars are shown below:
80, 80, 95, 100, 210
(b) Which of the averages best reflects the central tendency of the maximum speeds of the cars? Explain your answer briefly.
Solution
(b) Since there is an extreme value (210 km/h), the arithmetic
mean is not a suitable average.
Since the mode (80 km/h) is smaller than all the other data, it is also not a suitable average.
The median (95 km/h) is the best average to reflect the central tendency of the maximum speeds of the cars.

10. Comparing Two Sets of Data with Different Types of Average
The following shows the daily temperatures (in C) of city A and city B in a week.
City A: 0, 1, 5, 8, 8, 9, 11
City B: 7, 8, 9, 10, 11, 11, 14

11. City A: 0, 1, 5, 8, 8, 9, 11
City B: 7, 8, 9, 10, 11, 11, 14
I think city B was warmer because the mode of the temperatures of city B was higher than that of city A.
Do you agree with Amy’s explanation? No

12. We can conclude that city B was warmer than city A because
City A: arithmetic mean = 6 C , median = 8 C, mode = 8 C
City B: arithmetic mean = 10 C, median = 10 C, mode = 11 C
We can conclude that city B was warmer than city A because
the arithmetic mean, median and mode of the temperatures of city B are all higher than those of city A.

13. Follow-up question
The following are the heights (in cm) of the players in two basketball teams.
Team A: 170, 170, 173, 180, 181
Team B: 150, 161, 170, 170, 190
(a) Find (i) the mean height, (ii) the median height, (iii) the mode of the heights of the players in each team.
(b) Comparing the averages obtained in (a), which team’s players are taller? Explain your answer briefly.

14. Follow-up question (cont’d)
Team A: 170, 170, 173, 180, 181
Team B: 150, 161, 170, 170, 190
Solution
(a) (i) Mean height of the players in Team A 5
181
180
173
170 + = cm cm
174.8 =
Mean height of the players in Team B 5
190
170
161
150 + = cm cm
168.2 =

15. Follow-up question (cont’d)
Team A: 170, 170, 173, 180, 181
Team B: 150, 161, 170, 170, 190
Solution
(a) (ii) Median height of the players in Team A cm
173 =
Median height of the players in Team B cm
170 =
(a) (iii) Mode of the heights of the players in Team A cm
170 =
Mode of the heights of the players in Team B cm
170 =
(b) Team A. This is because the arithmetic mean and the median of the heights of the players in Team A are greater than those in Team B.

16. Misuses of Averages

17. Misuses of Averages
Sometimes, people may misuse averages to mislead the readers on purpose. Therefore, we should note the following points carefully.
whether the type of average used is appropriate,
which data are involved in the calculation.

18. Whether the type of average used is appropriate
for example:
I got 60, 60, 42, 38, 31 marks in 5 tests. So, my average score is 60.
Harry
It is not reasonable to use the mode as Harry’s average score. 60 is his highest score but his mean score is only 46.2.

19. Which data are involved in the calculation
For example: The members of a race team can finish 100 m race in 11.2 s, 12.4 s, 12.8 s, 13.6 s and 19.7 s respectively (mean = s).
After the slowest member left the team, the coach claimed that:
Due to my supervision, the mean race time of the team is changed from s to 12.5 s. It’s a significant improvement.

20. Which data are involved in the calculation
For example: The members of a race team can finish 100 m race in 11.2 s, 12.4 s, 12.8 s, 13.6 s and 19.7 s respectively (mean = s).
After the slowest member left the club, the coach claimed that:
Due to my supervision, the mean race time of the team is changed from s to 12.5 s. It’s a significant improvement.
The coach’s claim is not reasonable as the numbers of members involved in the calculations are not the same, and their results are not comparable.

21. Follow-up question
A computer company offers 5 different business programs for the customers. The following are the prices of the programs: $1499, $1499, $2799, $2999, $3399 The manager of the company claims that the average price of the programs is below $1500. Do you agree with his claim? Why?
Solution
No. As 3 out of 5 kinds of programs cost above $1500, it is not reasonable to use the mode, $ 1499, as the average price.