How to Find the Mean, Median and Mode for Discrete Data
Understanding how to calculate averages for discrete data is a key requirement in IGCSE Statistics. This page focuses on how to find the mean, median and mode for a list of discrete values, using the methods and presentation expected in exam questions.
Discrete data consists of values that can only take specific, separate numbers (things you count rather than measure), such as the number of siblings in a family or goals scored in a match. Once you have a set of discrete data, three different measures can each summarise its "centre": the mean, the median and the mode. Choosing the right one depends on the shape of your data and whether any extreme values are present.
Mean
The mean is calculated by adding together all the values in a data set and then dividing by how many values there are. Because every value contributes to the calculation, the mean makes full use of the data, but this also means a single unusually large or small value can pull it away from what feels like a "typical" result.
💡 Example 1: Straightforward list
Find the mean of: 5, 9, 6, 8, 7.
Add the values:
\[ 5 + 9 + 6 + 8 + 7 = 35 \]Divide by the count:
\[ \text{Mean} = \frac{35}{5} = 7 \]💡 Example 2: Working backwards from the mean
Six values have a mean of 12. Five of them are 8, 14, 10, 13 and 11. Find the sixth value.
Find the required total:
\[ \text{Total} = 12 \times 6 = 72 \]Subtract the known values:
\[\begin{aligned} 72 - (8 + 14 + 10 + 13 + 11) &= 72 - 56 \\ &= 16 \end{aligned}\]The sixth value is 16.
Median
The median is the middle value once the data has been written in ascending (smallest to largest) order. Because it sits in the centre of the ordered list rather than using the actual size of every value, extreme values do not affect it, making it a more reliable summary when outliers are present.
📌 Finding the position of the median
For \(n\) values arranged in order, the median is at position \(\dfrac{n+1}{2}\).
- If \(n\) is odd, this gives a whole number, so that single value is the median.
- If \(n\) is even, this lands between two values; the median is their mean.
💡 Example 3: Odd number of values
Find the median of: 11, 3, 7, 5, 9, 1, 6.
Write in ascending order:
1, 3, 5, 6, 7, 9, 11
Locate the middle (\(n = 7\)):
\[ \text{Position} = \tfrac{7+1}{2} = 4\text{th value} \]Median = 6
💡 Example 4: Even number of values
Find the median of: 4, 12, 7, 3, 9, 14.
Write in ascending order:
3, 4, 7, 9, 12, 14
Locate the middle (\(n = 6\)):
Position = \(\tfrac{6+1}{2} = 3.5\text{th}\)
Take the mean of the 3rd and 4th values:
\[ \text{Median} = \frac{7 + 9}{2} = 8 \]Mode
The mode is the value that appears most often in a data set. Unlike the mean and median, the mode can be used with non-numerical (categorical) data; for example, you can find the mode of a list of colours or shoe sizes. A data set may have no mode, one mode, or more than one mode.
No mode
Data: 2, 5, 8, 11
Every value appears exactly once, so there is no mode.
One mode
Data: 3, 5, 5, 7, 9
5 appears twice, more than any other value.
Mode = 5
Two modes (bimodal)
Data: 4, 4, 6, 9, 9, 11
Both 4 and 9 appear twice.
Modes = 4 and 9
When to Use Each Average
Each average has strengths and weaknesses. The table below summarises when each one is most appropriate; this comparison often appears directly in IGCSE exam questions.
| Average | Advantage | Disadvantage | Best used when... |
|---|---|---|---|
| Mean | Uses every value in the data set | Distorted by outliers (extreme values) | Data is spread fairly symmetrically with no extreme values |
| Median | Not affected by outliers | Does not use all the data | There are a few very high or very low values |
| Mode | Can be used for non-numerical data; not affected by outliers | May not exist; can be unrepresentative | You need the most typical or most popular value, or data is categorical |
💡 Example 5: Choosing the best average
Seven people donate to a charity collection: £8, £11, £13, £15, £17, £19, £250.
Mean:
\[\begin{aligned} \text{Mean} &= \frac{8+11+13+15+17+19+250}{7} \\[4pt] &= \frac{333}{7} \approx £47.57 \end{aligned}\]This is higher than six out of seven donations, distorted by the £250 outlier.
Median:
Ordered list: £8, £11, £13, £15, £17, £19, £250
Median = £15
This is much more representative of a typical donation.
The median is the better average here because the outlier (£250) does not affect it.
🔑 Key Points
- Always sort data into ascending order before finding the median.
- For an even number of values, the median is the mean of the two middle values.
- The mean, median and mode are not always values that appear in the original data set (mean and median often are not).
- A data set can have no mode, one mode, or more than one mode.
- The mode is the only average that can be used for categorical (non-numerical) data.
- If asked to find the mean without a calculator, check whether the sum divides exactly or requires a fraction/decimal answer.
⚠️ Common Mistakes
- Forgetting to order the list before finding the median; always sort first.
- Picking the middle index instead of the middle value: if there are 7 values, the median is the 4th value, not the number 7.
- Assuming the mode must exist: if all values appear once, there is no mode.
- Using the mean when there is an outlier: choose the median in those situations.
- Dividing by the wrong \(n\) when working backwards from the mean; multiply by \(n\) first, then subtract.