Comparing Different Types of Average

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.

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.

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.

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.

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   

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

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 =

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.

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

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

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.

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.

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 =

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.

Misuses of Averages

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.

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.

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 = 13.94 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 13.94 s to 12.5 s. It’s a significant improvement.

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 = 13.94 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 13.94 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.

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.