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This lesson covers DfE content statements L2.23, L2.24 and L2.25 — calculating mean, median, mode and range; finding the mean from a grouped frequency table; and using averages and range to compare two data sets.
In everyday life and the workplace, you need to summarise data quickly. "On average, how long does a delivery take?" "What is the typical salary?" Averages give a single representative value for a set of data.
| Measure | Definition | How to find it |
|---|---|---|
| Mean | The total divided by the number of values | Add all values, divide by count |
| Median | The middle value when ordered | Order the data, find the middle |
| Mode | The most common value | Find the value that appears most often |
| Range | The spread of the data | Highest value − Lowest value |
Mean = Sum of all values ÷ Number of values
Scenario: A shop records daily takings for a week: £420, £385, £510, £470, £395, £620, £540. What is the mean daily taking?
Sum = 420 + 385 + 510 + 470 + 395 + 620 + 540 = £3,340 Mean = £3,340 ÷ 7 = £477.14 (to 2 d.p.)
Scenario: A student's test scores are 72, 85, 68, 91, and 74. What mean score do they need on a sixth test to achieve an overall mean of 80?
Required total = 80 × 6 = 480 Current total = 72 + 85 + 68 + 91 + 74 = 390 Score needed = 480 − 390 = 90
Exam Tip: Questions that give you the mean and ask for a missing value are very common. Use the formula: Sum = Mean × Number of values. Then work backwards.
The median is the middle value when data is arranged in order.
Position of the median: (n + 1) ÷ 2, where n is the number of values.
Scenario: Five houses sold for: £185,000, £220,000, £195,000, £480,000, £205,000. Find the median.
Ordered: £185,000, £195,000, £205,000, £220,000, £480,000
Median = £205,000
Notice that the mean would be £257,000 — pulled up by the one expensive house. The median gives a better representation of the "typical" house price.
Scenario: Six employees have years of service: 2, 5, 7, 9, 12, 15. Find the median.
Position = (6 + 1) ÷ 2 = 3.5th value (between 3rd and 4th) Median = (7 + 9) ÷ 2 = 8 years
The mode is the value that appears most frequently. There can be no mode, one mode, or more than one mode.
Scenario: A shoe shop records sizes sold in one day: 6, 7, 8, 7, 9, 8, 7, 10, 8, 7, 6, 8, 7.
| Size | Frequency |
|---|---|
| 6 | 2 |
| 7 | 5 |
| 8 | 4 |
| 9 | 1 |
| 10 | 1 |
Mode = Size 7 (appears 5 times)
This helps the shop know which size to stock the most of.
Range = Highest value − Lowest value
The range measures the spread of the data. A larger range means more variation.
Scenario: Daily maximum temperatures for a week: 18°C, 21°C, 16°C, 23°C, 19°C, 24°C, 17°C.
Range = 24 − 16 = 8°C
| Average | Best used when... | Weakness |
|---|---|---|
| Mean | Data has no extreme values; you want to use all the data | Distorted by outliers |
| Median | Data has extreme values (outliers); you want the "middle" | Ignores extreme values |
| Mode | Data is categorical (e.g. favourite colour, shoe size) | May not exist; may not be central |
Scenario: A company has 10 employees with these salaries: £22k, £24k, £25k, £25k, £26k, £27k, £28k, £30k, £35k, £120k.
Mean = £362k ÷ 10 = £36.2k (distorted by the £120k salary) Median = (£26k + £27k) ÷ 2 = £26.5k (better representation) Mode = £25k (appears twice)
A recruitment advert claiming "average salary £36.2k" would be misleading. The median of £26.5k is more representative.
When data is presented in a frequency table, use:
Mean = Sum of (value × frequency) ÷ Sum of frequencies
Scenario: Customers rate a service from 1 to 5:
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