6.7 Choosing the best average
For some sets of data it may be better to use one type of average over another as it will be more representative of the data type.
Here are some of the advantages and disadvantages of each type of average.
Mean
Advantages
Uses all the data values.
Disadvantages
May not always be one of the values in the set of data or a value that does not make sense for the data, e.g. 1.6 people.
- The mean may not be useful for a set of data which has a value a lot higher or lower than the others, e.g. If you were to include the salary of the Managing Director with the wages of shop floor staff when calculating an average salary, it is likely that the mean would be distorted by the higher salary of the Managing Director, which means that it would not represent the data very well.
Median
Advantages
It is the middle value so is not affected by very high or very low values. It is a useful type of average for data sets with such values.
Disadvantages
Sometimes it will not be one of the values in the data set.
Mode
Advantages
It will always be one of the values in the data set (if there is a mode).
It is very good for certain types of data, e.g. finding the most common shoe size.
Disadvantages
There may not be a mode.
There may be several modes.
It may be at one end of the data distribution.
Now have a go at finding the mean, median and mode.
Activity 16: Finding different averages
A bridal shop records the sizes of wedding dresses that it sells in one month. The table below shows the results.
Find the following:
- a.the mean
- b.the median
- c.the mode.
Which of the averages gives the most useful information for this data set?
| Dress size | No. of dresses sold |
|---|---|
| 8 | 2 |
| 10 | 8 |
| 12 | 11 |
| 14 | 12 |
| 16 | 5 |
| 18 | 2 |
Answer
(a) The mean
First, work out the total number of dresses sold by multiplying the dress size column by the number of dresses sold.
Add a frequency column to display your calculations.
Next, work out the totals for the number of dresses sold column and the frequency column.
| Dress size | No. of dresses sold | Frequency |
| 8 | 2 | 8 × 2 = 16 |
| 10 | 8 | 10 × 8 = 80 |
| 12 | 11 | 12 × 11 = 132 |
| 14 | 12 | 14 × 12 = 168 |
| 16 | 5 | 16 × 5 = 80 |
| 18 | 2 | 18 × 2 = 36 |
| Totals | Highlighted40 | Highlighted512 |
Finally calculate the mean by dividing the frequency by the number of dresses sold:
512 ÷ 40 = 12.8
Mean = 12.8
- (b) The median
First you need to calculate the total number of dresses sold. To do this add up each number in the frequency table:
2 + 8 + 10 + 13 + 5 + 2 = 40 dresses sold.
Now find the middle value by listing the quantity of each size of dress from smallest to biggest:
8 8 10 10 10 10 10 10 10 10 12 12 12 12 12 12 12 12 12 12 12 14 14 14 14 14 14 14 14 14 14 14 14 16 16 16 16 16 18 18
The midpoint of the 40 dresses sold is between values 20 and 21 so you need to find both of these values. In this example they are 12 and 12.
You calculate the median by adding 12 + 12 and dividing by 2:
12 + 12 = 24
24 ÷ 2 = 12
As you can see, in this example the median is size 12.
A quicker way to do this is to calculate the number of dresses, which you know is 40.
You then use the frequency table to find the midpoint which is between the 20th and the 21st dress size. If you count up the number of dresses sold in the frequency table:
2 + 8 + 11 = 21
you can see that the 20th and 21st values fall within size 12, so the median is size 12.
Median = 12
(c) The mode
To find the mode you need to look for the highest number of dresses sold. In this case it is 12 which shows that the mode dress size is 14.
Mode = 14
| Dress size | No. of dresses sold |
|---|---|
| 8 | 2 |
| 10 | 8 |
| 12 | 11 |
| Highlighted14 | Highlighted12 |
| 16 | 5 |
| 18 | 2 |
In this case the mean is size 12.8 which does not exist as a dress size so this is not useful.
The median result is size 12 which is a dress size, but it is still not the most commonly sold dress size.
The mode is dress size 14 which is the most commonly sold dress size and so this gives the most useful information.
Well done! You have now learned all you need to know about mean, median, mode and range. The final part of this section, before the end-of-course quiz, looks at probability.
Summary
In this section you have learned:
- that there are different types of averages that can be used when working with a set of data – mean, median and mode
- range is the difference between the largest data value and the smallest data value and is useful for comparing how consistently someone or something performs
- mean is what is commonly referred to when talking about the average of a data set
- how to find the mean from both a single data set and also a set of grouped data
- what the median of a data set is and how to find it for a given set of data
- what the mode of a data set is and how to find it for a given set of data
- how to choose the ‘best’ type of average for a given set of data.