It is often very useful to determine the centre of a data distribution to be able to compare the data to other sets, or to summarise key features of the data. Three measures of centre are the mean (average), median and the mode.
Mean
The mean is often referred to as the average. To calculate the mean, add all the scores in a data set, then divide this by number of scores.
The mean of a data set, denoted $\overline{x}$x, is given by:
$\overline{x}=\frac{\sum_{\quad}^{\quad}\ x}{n}$x= ∑ xn
Practice questions
Question 1
The mean of $4$4 scores is $21$21. If three of the scores are $17$17, $3$3 and $8$8, find the $4$4th score, $x$x.
Enter each line of working as an equation.
Median
The median is the middle score in a data set, when the data is sorted to be in order. If there are two middle scores, we take the average of them to find the median.
To find the median of a data set:
- Sort the data points into order
- If there are an odd number of scores, the median is the middle score (the $\frac{n+1}{2}$n+12th score)
- If there are an even number of scores, the median is calculated by finding the mean of the two middle scores (the $\frac{n}{2}$n2th and $\left(\frac{n}{2}+1\right)$(n2+1)th scores)
Worked example
Find the median of the nine numbers below:
$1,1,3,5,7,9,9,10,15$1,1,3,5,7,9,9,10,15
Think: This data set is already sorted in ascending order. There are $9$9 scores in total, which is an odd number, so the median will be the score in the middle.
Do:
| $\text{middle term position}$middle term position | $=$= | $\frac{9+1}{2}$9+12 |
| $=$= | $5$5th |
This means that the fifth term will be the median: $1,1,3,5,\editable{7},9,9,10,15$1,1,3,5,7,9,9,10,15, which is $7$7.
Reflect: This data set had an odd number of scores, so the median was just the middle score. Now consider this data set, which has four scores: $8,12,17,20$8,12,17,20.
There are two middle scores for this set - the second and third scores - and so the median will be the mean of the second and third terms.
Median $=$=$\frac{12+17}{2}=14.5$12+172=14.5
Practice questions
Question 2
Find the median from the frequency distribution table:
| Score | Frequency |
|---|---|
| $23$23 | $2$2 |
| $24$24 | $26$26 |
| $25$25 | $37$37 |
| $26$26 | $24$24 |
| $27$27 | $25$25 |
Mode
The mode describes the most frequently occurring score.
Suppose that $10$10 people were asked how many pets they had. $2$2 people said they didn't own any pets, $6$6 people had one pet and $2$2 people said they had two pets.
In this data set, the most common number of pets that people have is one pet, and so the mode of this data set is $1$1.
A data set can have more than one mode, if two or more scores are equally tied as the most frequently occurring.
The mode of a data set is the most frequently occurring score.
Practice question
Question 3
Find the mode of the following scores:
$8,18,5,2,2,10,8,5,14,14,8,8,10,18,14,5$8,18,5,2,2,10,8,5,14,14,8,8,10,18,14,5
Mode = $\editable{}$