Finding the median from a frequency table
Here are Kieran’s results in a frequency table
| Number of tracks on album | Frequency |
| 9 | 1 |
| 10 | 4 |
| 11 | 3 |
| 12 | 3 |
| 13 | 0 |
| 14 | 1 |
| Number of tracks on album | 9 |
|---|---|
| Frequency | 1 |
| Number of tracks on album | 10 |
|---|---|
| Frequency | 4 |
| Number of tracks on album | 11 |
|---|---|
| Frequency | 3 |
| Number of tracks on album | 12 |
|---|---|
| Frequency | 3 |
| Number of tracks on album | 13 |
|---|---|
| Frequency | 0 |
| Number of tracks on album | 14 |
|---|---|
| Frequency | 1 |
From the table, we can find the total number of albums
\(1 + 4 + 3 + 3 + 0 + 1 = 12\, albums\)
As the results are in a table, they are already ordered for us. In this case, as there are \(12\) results, the median is between the \(6th\) and \(7th\) result.
We look at the frequency column and determine when the cumulative frequency passes the \(6th\) and \(7th\) results.
(There is an explanation of Cumulative Frequency on a previous page in this section)
The first row has cumulative frequency \(1\)
The second row has cumulative frequency \(5\)
The third row has cumulative frequency \(8\)
The third row passes the \(6th\) and \(7th\) results, so the median must be in there.
Median is \(11\) tracks
In general, if there are \(n\) results, the median will be result \((n+1) \div 2\)
For example: for five numbers, the median is the \((5+1) \div 2 = 3rd\) result.
For six numbers, the median is the result between \(3rd\) and \(4th\) due to the fact that \((6 + 1) ÷ 2 = 3.5\).
Question
Find the median number of tracks on Suzie's albums.
| Number of tracks on album | Frequency |
| 6 | 3 |
| 7 | 0 |
| 8 | 2 |
| 9 | 1 |
| 10 | 3 |
| 11 | 4 |
| 12 | 2 |
| Number of tracks on album | 6 |
|---|---|
| Frequency | 3 |
| Number of tracks on album | 7 |
|---|---|
| Frequency | 0 |
| Number of tracks on album | 8 |
|---|---|
| Frequency | 2 |
| Number of tracks on album | 9 |
|---|---|
| Frequency | 1 |
| Number of tracks on album | 10 |
|---|---|
| Frequency | 3 |
| Number of tracks on album | 11 |
|---|---|
| Frequency | 4 |
| Number of tracks on album | 12 |
|---|---|
| Frequency | 2 |
Suzie has \(15\) albums, so the median is the \(8th\) result (Remember we can use \((15 + 1) \div 2 \))
The cumulative frequency passes the eighth album at the fifth row.
The median is \(10\).
Remember, when you are working out the median:
- Put the results in numerical order (in a frequency table this will already be done)
- Count the total amount of results and add one
- Divide this by 2 to find the the position of the middle result
- Find the middle result in the numerically ordered list or frequency table
- You will then have the median of the set of results
