These questions describe a small frequency table in sentences, then ask for a total, a total-based average, an ordered centre, the most frequent value, endpoint spread or a missing frequency. Keep each recorded value paired with the number of times it occurs.
Keep value and frequency in separate columns
Rewrite the description mentally as pairs. Suppose the value 2 occurs twice, 5 occurs three times and 8 occurs once. The frequencies add to 6, so there are six observations. The values total 2 × 2 + 5 × 3 + 8 × 1 = 27, so the total-based average is 27 divided by 6, or 4.5. Multiplication matters because each value contributes once for every occurrence.
| Quantity wanted | Reliable method |
|---|---|
| Number of observations | Add the frequencies. |
| Total-based average | Add value × frequency products; divide that total by the sum of the frequencies. |
| Most frequent value | Find the greatest frequency, then report the value paired with it. |
| Ordered centre | Find the middle position or two middle positions and count through the frequency runs until you reach them. |
| Endpoint spread | Subtract the smallest occurring value from the largest occurring value. |
| Missing frequency | Subtract the known frequencies from the stated total number of observations. |
A frequency of zero contributes neither an observation nor anything to the total, even if its value is mentioned. For the ordered centre, frequencies control positions: in a set with eleven observations, locate the sixth observation by moving through the runs. You usually do not need to write the expanded list.
Where mistakes come from
| Tempting route | How to reject it |
|---|---|
| Add the named values to count the observations | Named values describe results, not people or measurements. Only the frequencies count observations. |
| Report the greatest frequency instead of its value | The frequency says how often; report the value attached to that greatest frequency. Zero can be the result when it occurs most often. |
| Add values and frequencies together for the data total | A frequency repeats its value. Multiply each pair first, then add the products. |
| Average only the different values that are named | That gives every named value one vote. The total-based average must give a value as many contributions as its frequency states. |
| Take the middle named value as the ordered centre | The labels are not the expanded data. Locate the middle observation by counting through their frequencies. |
| Use frequencies or merely the largest value for the range | Range is a difference between the two end values that actually occur; frequencies do not enter the subtraction. |
Try it
A record shows the value 1 four times, the value 2 twice and the value 3 four times. What is the median of these ten values?
- 2
- 1.5
- 2.5
- 1
A table records the value 1 three times, the value 4 twice and the value 5 three times. What is the mean of these eight values?
- 8.67, from dividing 26 by the 3 values that are named
- 3.33
- 26, the total of all eight values
- 3.25
A frequency table records 20 pupils in total. It shows 7 pupils scoring 1 mark, 5 pupils scoring 2 marks and the rest scoring 3 marks. How many pupils scored 3 marks? Enter the number of pupils.