Mean, median and mode

Notation

For raw data x1, x2, ..., xn the mean is x̄ = Σx / n. For a frequency table it is x̄ = Σfx / Σf. The median is the middle value of the ordered data; the mode is the most common value (the modal class for grouped data).

From a frequency table

Goals in 24 matches: Σf = 24, Σfx = 0 + 8 + 12 + 9 + 8 = 37, so x̄ = 37/24 = 1.54 (3 s.f.). The median is the 12.5th value: the cumulative frequencies are 5, 13, ..., so the 12th and 13th values are both 1 and the median is 1. The mode is 1.

Choosing a measure

• Mean: uses every value; best for symmetrical data without outliers.
• Median: not affected by outliers or skew; best for skewed data.
• Mode: the only one for qualitative data; can be useful for discrete data.

Combining groups: add the totals, not the means. Total = n × mean for each group.

Worked example

Class A (20 students) has mean 64 and class B (30 students) has mean 58. Find the mean of all 50 students.

1. Total for A = 20 × 64 = 1280; total for B = 30 × 58 = 1740.
2. Combined total = 3020; divide by 50.

Answer: Mean = 60.4

Check yourself

1. Find the mean number of goals per match from the table. Give your answer to 3 significant figures.
   Show answer

   1.54. Σfx = 37, Σf = 24, so x̄ = 37/24 = 1.54.

2. Find the median number of goals from the table.
   Show answer

   1. 24 values: the median is the mean of the 12th and 13th. Cumulative frequencies 5, 13: both are 1.

3. For 15 values, Σx = 408. Find the mean. Give your answer to 3 significant figures.
   Show answer

   27.2. x̄ = 408/15 = 27.2.

4. Class A (20 students) has mean 64 and class B (30 students) has mean 58. Find the mean of all 50 students.
   Show answer

   60.4. (1280 + 1740)/50 = 60.4.

5. The mean of 9 values is 12. A tenth value, 22, is added. Find the new mean.
   Show answer

   13. New total = 108 + 22 = 130, so the mean is 130/10 = 13.

6. House prices in a town include a few very expensive houses. Which average best describes a typical price?
   • The mean, because it uses every value
   • The mode, because prices are qualitative
   • The median, because it is not affected by the extreme values
   • The range, because it shows the extremes
   Show answer

   The median, because it is not affected by the extreme values. Extreme values pull the mean but not the median.

7. Which measure of location can be used for the favourite colours of a group of students?
   • The mode
   • The mean
   • The median
   • The interquartile range
   Show answer

   The mode. Colours are qualitative: they cannot be added or ordered, so only the mode works.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S1

Last updated 7 November 2026. Spotted a mistake? Tell us.