Describing data: mean, median, mode and standard deviation

Averages

• Mean: sum ÷ number. Uses all data but is pulled by extreme values.
• Median: middle value when data are in order (mean of the middle two for an even number). Not affected by an extreme value.
• Mode: most common value. Useful for categoric or discrete data.

In a normal distribution, mean, median and mode are equal.

Standard deviation

s = √[ Σ(x - x̄)² / (n - 1) ]

• About 68% of normally distributed values lie within x̄ ± 1s and about 95% within x̄ ± 2s.
• s has the same units as the data.
• s² (the variance) appears in the t-test: s² / n for each group.
• Error bars of ± 1s on a bar chart show variability; if they overlap a lot, a significant difference is unlikely.

On a calculator, the sample standard deviation key is usually labelled sₓ or σₙ₋₁; check it gives the same answer as the formula.

Range bars and interquartile thinking

Some questions ask for range bars instead (the October 2023 mark scheme: 'range bars plotted correctly' from the lowest to the highest value). Range is quick but uses only two values; standard deviation is more informative because it uses every value.

Worked example

Calculate the mean, median and standard deviation of the heart rates (beats per 10 s) 24, 27, 25, 30, 26, 24.

1. Mean = 156 / 6 = 26.0
2. Ordered: 24, 24, 25, 26, 27, 30; median = (25 + 26) / 2 = 25.5
3. Σ(x - x̄)² = 26; ÷ 5 = 5.2
4. s = 2.28

Answer: Mean 26.0, median 25.5, mode 24, s = 2.28 beats per 10 s

Check yourself

1. Find the median of 14, 9, 11, 20, 12.
   Show answer

   12. Ordered: 9, 11, 12, 14, 20; middle value 12.

2. Find the median of 3.1, 2.8, 3.6, 3.3.
   Show answer

   3.2. Ordered 2.8, 3.1, 3.3, 3.6; (3.1 + 3.3) / 2 = 3.2.

3. Calculate the standard deviation of 5, 7, 9, 11, 13 to 3 significant figures.
   Show answer

   3.16. Mean 9; squares 16, 4, 0, 4, 16 = 40; 40 / 4 = 10; √10 = 3.16.

4. A group of 5 values has a standard deviation of 0.40. Calculate s² / n, which is used in the t-test.
   Show answer

   0.032. 0.40² = 0.16; 0.16 / 5 = 0.032.

5. Which average is least affected by one very large value?
   • Mean
   • Sum
   • Range
   • Median
   Show answer

   Median. The median depends only on the middle position.

6. Roughly what percentage of normally distributed data lie within one standard deviation of the mean?
   • 50%
   • 68%
   • 95%
   • 100%
   Show answer

   68%. About 68% within ±1s, about 95% within ±2s.

7. Lengths are measured in mm. What are the units of their standard deviation?
   • mm²
   • No units
   • mm⁻¹
   • mm
   Show answer

   mm. s has the same units as the data; the variance would be mm².

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Sources

• Pearson Edexcel International Advanced Level in Biology specification (XBI11/YBI11, first teaching 2018): Unit 3, Unit 6 and Appendix 6 mathematical skills
• Pearson mark scheme WBI16/01, October 2023 (via Physics & Maths Tutor)

Last updated 8 November 2026. Spotted a mistake? Tell us.