Variability: range, standard deviation and error bars

Range and standard deviation

• Range = largest value - smallest value. Easy, but depends only on the two extremes.
• Standard deviation (s) measures the spread of all the data about the mean. A small s means data are close to the mean (repeatable, precise).

s = √[ Σ(x - x̄)² / (n - 1) ]

where x̄ is the mean and n the number of values. This is the sample standard deviation (divide by n - 1), the version Pearson uses. The formula is given in the paper when a calculation is needed, sometimes in the equivalent form s = √[ (Σx² - (Σx)²/n) / (n - 1) ]. On a calculator use the sₓ (σₙ₋₁) key, not σₓ (σₙ), which divides by n and gives a smaller answer.

About 68% of normally distributed data lie within one standard deviation of the mean, and about 95% within two.

Calculating s step by step

1. Find the mean x̄.

2. Subtract the mean from each value: (x - x̄).

3. Square each difference: (x - x̄)².

4. Add them: Σ(x - x̄)².

5. Divide by n - 1.

6. Square root.

Set out a table of x, (x - x̄) and (x - x̄)² to avoid slips.

Data 12, 15, 11, 14, 13: mean 13

Error bars and overlap

Plot mean ± 1 standard deviation (or range bars) as vertical bars on a graph.

• If error bars of two means overlap a lot, the difference may be due to chance: it is unlikely to be significant.
• If they do not overlap, the difference may be significant, but only a statistical test (Unit 6) can confirm it.

The June 2024 mark scheme: to show results are repeatable, repeat for every group, calculate means and standard deviations, and plot means with standard deviation bars.

Worked example

Calculate the standard deviation of 12, 15, 11, 14, 13.

1. Mean = 65 / 5 = 13
2. Squared differences: 1, 4, 4, 1, 0; sum = 10
3. Divide by n - 1 = 4: 2.5
4. s = √2.5 = 1.58

Answer: s = 1.58 (3 s.f.: 1.58)

Check yourself

1. Repeat readings are 2.3, 2.9, 2.6, 2.4 cm³. Calculate the range.
   Show answer

   0.6 cm³. 2.9 - 2.3 = 0.6 cm³.

2. Calculate the standard deviation of 4, 6, 8, 10 using s = √[Σ(x - x̄)² / (n - 1)]. Give your answer to 3 significant figures.
   Show answer

   2.58. Mean 7; squared differences 9, 1, 1, 9 = 20; 20 / 3 = 6.67; √6.67 = 2.58.

3. Zones of inhibition were 18, 20, 19, 21 and 17 mm. Calculate the standard deviation to 3 significant figures.
   Show answer

   1.58 mm. Mean 19; squared differences 1, 1, 0, 4, 4 = 10; 10 / 4 = 2.5; s = 1.58 mm.

4. For the data 3, 5, 7, what is Σ(x - x̄)²?
   Show answer

   8. Mean 5; (−2)² + 0² + 2² = 8.

5. Two groups have the same mean, but group A has a much smaller standard deviation. What does this show?
   • Group A's mean is more accurate
   • Group A's values are closer to the mean, so more repeatable
   • Group A has fewer values
   • Group B's results are wrong
   Show answer

   Group A's values are closer to the mean, so more repeatable. Smaller s means less spread.

6. Error bars (mean ± 1 SD) for two treatments overlap a lot. What is the best conclusion?
   • The treatments definitely have the same effect
   • The difference is definitely significant
   • The experiment has no anomalies
   • The difference between the means may be due to chance
   Show answer

   The difference between the means may be due to chance. Overlap suggests no significant difference, but a test is needed to be sure.

7. A question asks how to show that results are repeatable. Which answer is best?
   • Use a different method
   • Repeat each treatment several times, calculate means and standard deviations, and plot means with SD bars
   • Record results to more decimal places
   • Do the experiment once at a higher temperature
   Show answer

   Repeat each treatment several times, calculate means and standard deviations, and plot means with SD bars. This matches the June 2024 mark scheme.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Biology specification (XBI11/YBI11, first teaching 2018): Unit 3, Unit 6 and Appendix 6 mathematical skills
• Pearson Examiners' Report WBI13/01, June 2024
• Pearson mark scheme WBI13/01, June 2024 (via Physics & Maths Tutor)

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