Averages and spread from a list
Univariate data records just one variable, such as a set of test scores or shoe sizes. An average is a single value that stands in for the whole set, and there are three of them. The range sits alongside them to describe how spread out the data is.
- Mean
- Add every value, then divide by how many values there are. The "fair share" average.
- Median
- Put the values in order and take the middle one. With an even count, take the mean of the two middle values.
- Mode
- The value that appears most often. A set can have one mode, several, or none.
- Range
- The largest value minus the smallest. It measures spread, not a central value.
Written as a formula, the mean of \(n\) values is:
Here \(\bar{x}\) is read "x-bar" and \(\sum x\) means "the sum of all the values".
Six matches produced these goals: 1, 3, 3, 0, 5, 2. Find the mean, median, mode and range.
Order before you find the median
The most common slip is taking the middle of the unordered list. Sorting the values first takes a few seconds and removes the mistake entirely.
Averages from a frequency table
When the same values repeat a lot, data is often given as a frequency table: each value \(x\) with the number of times it occurs, its frequency \(f\). You do not need to write the list out in full. Instead, multiply each value by its frequency, total those products, and divide by the total frequency:
The table shows the number of pets owned by 20 pupils. Find the mean, median, mode and range.
| Pets \(x\) | Frequency \(f\) | \(f x\) |
|---|---|---|
| 0 | 5 | 0 |
| 1 | 8 | 8 |
| 2 | 4 | 8 |
| 3 | 3 | 9 |
| Total | 20 | 25 |
Mean: \(\bar{x} = \dfrac{\sum f x}{\sum f} = \dfrac{25}{20} = 1.25\) pets.
Median: there are 20 values, so the median is the mean of the 10th and 11th. Counting up the frequencies, the first 5 values are 0 and the next 8 (positions 6 to 13) are 1, so both the 10th and 11th values are \(1\). The median is \(1\).
Mode: the highest frequency is 8, at \(x = 1\), so the mode is \(1\). Range: \(3 - 0 = 3\).
Where this is assessed
Calculating averages from a table is Criterion A (knowing the method) and applying it to context is Criterion D. Laying out a neat \(f x\) column with clear totals is exactly the sort of organised working that Criterion C rewards.
Which average, and when
Part of the skill is picking the right summary for the data in front of you, not just calculating all three.
- Mean: uses every value, so it is the most informative, but a single very large or very small value drags it off.
- Median: ignores the exact size of extreme values, so it is the safer average when the data has outliers, for example house prices or salaries.
- Mode: the only average that works for non-numerical data, such as the most common shoe size or favourite colour.
The mode is a value, not a frequency
If the size 6 shoe is the most common, the mode is 6, not the number of people who wear it. Reporting the frequency instead of the value is a classic mark loser.
Check yourself
Work each one through before opening the answer.
1. Find the mean, median, mode and range of: 3, 7, 7, 2, 6. +
Sum \(= 3+7+7+2+6 = 25\), so the mean \(= \dfrac{25}{5} = 5\). Ordered: \(2, 3, 6, 7, 7\), so the median (middle of five) is \(6\). The mode is \(7\) (it appears twice). Range \(= 7 - 2 = 5\). So mean \(= 5\), median \(= 6\), mode \(= 7\), range \(= 5\).
2. A frequency table gives shoe sizes: size 4 with \(f = 2\), size 5 with \(f = 6\), size 6 with \(f = 7\), size 7 with \(f = 5\). Find the mode and median. +
Total frequency \(= 2 + 6 + 7 + 5 = 20\). The highest frequency is 7 at size 6, so the mode is 6. The median is the mean of the 10th and 11th values. Counting up: sizes 4 fill positions 1 to 2, size 5 fills 3 to 8, size 6 fills 9 to 15, so both the 10th and 11th values are size 6. The median is 6.
3. A company's salaries include one very high figure for the director. Which average best describes a typical salary, and why? +
The median. The single very high director's salary would pull the mean upwards and make it unrepresentative, while the median simply takes the middle value and is not distorted by that one outlier.
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