Mean, Median, Mode — GCSE Maths

Calculate and interpret the three averages from data

What are averages?

  • An average is a single number that represents a set of data.
  • There are three types of average: mean, median and mode.
  • At GCSE, you need to: calculate all three, know when to use each, and interpret results.
  • You may also need to find averages from frequency tables.

Key Words

  • Mean — Add all values, divide by how many there are
  • Median — The middle value when data is in order
  • Mode — The most common value
  • Range — The difference between the largest and smallest values

Calculating the mean

  • Mean = sum of all values ÷ number of values.
  • Example: 4, 7, 9, 6, 4. Sum = 30. Count = 5. Mean = 30 ÷ 5 = 6.
  • The mean uses every value in the data set.
  • It can be affected by very large or small values (outliers).
  • Useful for comparing two groups with the same number of values.

Finding the median and mode

  • Median: arrange in order, find the middle. For even counts, average the two middle values.
  • Data: 3, 5, 7, 9, 11. Median = 7 (middle of 5 values).
  • Data: 2, 4, 6, 8. Median = (4 + 6) ÷ 2 = 5.
  • Mode: the value that appears most often.
  • Data: 2, 3, 3, 5, 7, 3. Mode = 3 (appears 3 times).

Choosing the right average

  • Mean: best for numerical data without extreme values.
  • Median: best when there are outliers (very high or low values).
  • Mode: best for categorical data (e.g. most popular colour).
  • Range tells you about the spread — a larger range = more variation.
  • In exams you may be asked to "compare" two data sets — use mean AND range.

Worked Examples

Example 1: Find all three averages

  1. Data: 8, 3, 7, 3, 5, 9, 3.
  2. Ordered: 3, 3, 3, 5, 7, 8, 9.
  3. Mean: (8+3+7+3+5+9+3) ÷ 7 = 38 ÷ 7 ≈ 5.4.
  4. Median: middle value = 5 (4th of 7).
  5. Mode: 3 (appears 3 times).

Example 2: Even number of values

  1. Data: 12, 15, 18, 21.
  2. Mean: (12+15+18+21) ÷ 4 = 66 ÷ 4 = 16.5.
  3. Median: (15 + 18) ÷ 2 = 16.5.
  4. Mode: none (all values appear once).
  5. Range: 21 − 12 = 9.

Practice Questions

  • Find the mean of: 5, 8, 11, 6, 10
    Answer: 8
    (5+8+11+6+10) = 40. 40 ÷ 5 = 8.
  • Find the median of: 3, 7, 2, 9, 5
    Answer: 5
    Ordered: 2, 3, 5, 7, 9. Middle value = 5.
  • Find the mode of: 4, 6, 4, 8, 6, 4, 9
    Answer: 4
    4 appears 3 times — more than any other value.
  • Find the median of: 10, 14, 18, 22
    Answer: 16
    (14 + 18) ÷ 2 = 32 ÷ 2 = 16.
  • Which average is least affected by an outlier?
    Answer: Median
    The median is not affected by extreme values — only the middle value matters.

Exam-Style Question

Seven students scored these marks in a test: 45, 52, 61, 52, 78, 90, 52. (a) Find the mean, median and mode. (b) Which average best represents the data? Give a reason.

Mark Scheme

  • 1 mark each: Mean = 430÷7 ≈ 61.4, Median = 52, Mode = 52
  • 1 mark: Mean best represents — it uses all values and is not distorted (accept median with justification)

Model Answer: Ordered: 45, 52, 52, 52, 61, 78, 90 Mean = (45+52+61+52+78+90+52)÷7 = 430÷7 ≈ 61.4 Median = 52 (4th value) Mode = 52 (appears 3 times) The mean best represents the data as it uses all values and gives a central view of performance.

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