Averages from tables
To make calculations more efficient, tables are often used to summarise the data.
The table below shows the goals scored in 10 football matches.
| Number of goals | Frequency |
| 0 | 2 |
| 1 | 2 |
| 2 | 5 |
| 3 | 1 |
| Number of goals | 0 |
|---|---|
| Frequency | 2 |
| Number of goals | 1 |
|---|---|
| Frequency | 2 |
| Number of goals | 2 |
|---|---|
| Frequency | 5 |
| Number of goals | 3 |
|---|---|
| Frequency | 1 |
Finding the mode from a table
The mode is the value which appears most often. When using a table, the mode is the value with the highest frequencyHow many times a value occurs..
Using the above table, there were 5 football matches where 2 goals were scored, which is a higher frequency than any other amount of goals.
The mode is 2. The modal number of goals scored is 2.
Finding the median from a table
The median is the value of the middle item of data when all the data is arranged in order. In the table above, the amounts of goals are already in order, as they start with zero goals and move up to three goals scored.
With a total frequency of 10 items of data, the median will be half way between the 5th and 6th items of data. Create a cumulative frequency (running total) column until you find the 5th and 6th items of data.
| Number of goals | Frequency | Cumulative frequency (running total) | |
| 0 | 2 | 2 | |
| 1 | 2 | \(2+2=4\) | |
| 2 | 5 | \(4+5=9\) | |
| 3 | 1 | \(9+1=10\) | |
| Total | 10 |
| Number of goals | 0 |
|---|---|
| Frequency | 2 |
| Cumulative frequency (running total) | 2 |
| Number of goals | 1 |
|---|---|
| Frequency | 2 |
| Cumulative frequency (running total) | \(2+2=4\) |
| Number of goals | 2 |
|---|---|
| Frequency | 5 |
| Cumulative frequency (running total) | \(4+5=9\) |
| Number of goals | 3 |
|---|---|
| Frequency | 1 |
| Cumulative frequency (running total) | \(9+1=10\) |
| Total | |
|---|---|
| Number of goals | |
| Frequency | 10 |
| Cumulative frequency (running total) | |
The third group starts with the 5th item of data and ends with the 9th item of data, so the 5th and 6th items of data will both be 2. The median number of goals is 2.
Finding the mean from a table
The mean is calculated by adding up all of the data and dividing by the number of items of data.
\(\text{mean} = \frac{\text{the sum of all the data}}{\text{the number of items of data}}\)
To find the mean in this example, the total number of goals must be found and then divided by the number of games.
From the table, it can be seen that in 2 games no goals were scored. This makes a grand total of zero goals so far. The rest of the total amount of goals can be worked out in this way, by multiplying goals (\(x\)) by the frequency (\(f\)). Call this column \(fx\) (\(f\) multiplied by \(x\)).
| Number of goals (\(x\)) | Frequency (\(f\)) | \(fx\) | |
| 0 | 2 | \(0 \times 2 = 0\) | |
| 1 | 2 | \(1 \times 2 = 2\) | |
| 2 | 5 | \(2 \times 5 = 10\) | |
| 3 | 1 | \(3 \times 1 = 3\) | |
| Total | 10 | 15 |
| Number of goals (\(x\)) | 0 |
|---|---|
| Frequency (\(f\)) | 2 |
| \(fx\) | \(0 \times 2 = 0\) |
| Number of goals (\(x\)) | 1 |
|---|---|
| Frequency (\(f\)) | 2 |
| \(fx\) | \(1 \times 2 = 2\) |
| Number of goals (\(x\)) | 2 |
|---|---|
| Frequency (\(f\)) | 5 |
| \(fx\) | \(2 \times 5 = 10\) |
| Number of goals (\(x\)) | 3 |
|---|---|
| Frequency (\(f\)) | 1 |
| \(fx\) | \(3 \times 1 = 3\) |
| Total | |
|---|---|
| Number of goals (\(x\)) | |
| Frequency (\(f\)) | 10 |
| \(fx\) | 15 |
The total number of goals is 15. There were 10 football games so \(15 \div 10 = 1.5\).
The mean number of goals is 1.5 goals per game.
Remember to divide\(fx\) by the total of the frequencies, not by the amount of different items of data 鈥 the correct answer here is \(\frac{15}{10}\) not \(\frac{15}{4}\).