Statistics · Variability
What is Standard Deviation?
The most widely used measure of spread — how it works, how to calculate it, and why it matters more than the range.
The mean tells you where the centre of your data is — but it tells you nothing about how spread out the data is. Two classes can have the same average exam score of 65 but very different distributions: one where every student scored between 60 and 70, and another where scores ranged from 20 to 100. The statistic that captures this spread is the standard deviation.
📘 Key Term
The standard deviation (σ or s) measures how much individual data values deviate from the mean on average. A low standard deviation means data points cluster closely around the mean; a high standard deviation means they are spread widely. It is always expressed in the same units as the original data.
Population vs Sample Standard Deviation
| Population (σ) | Sample (s) | |
|---|---|---|
| Formula | σ = √[Σ(x−μ)² / N] | s = √[Σ(x−x̄)² / (n−1)] |
| Denominator | N (all values) | n−1 (Bessel’s correction) |
| Used when | You have data on the entire population | You have a sample and want to estimate σ |
Step-by-Step Calculation
Dataset: 4, 7, 13, 2, 9 (n = 5)
Step 1: Calculate the mean: x̄ = (4+7+13+2+9)/5 = 35/5 = 7
Step 2: Find deviations from mean: −3, 0, +6, −5, +2
Step 3: Square each deviation: 9, 0, 36, 25, 4
Step 4: Sum of squares = 74
Step 5: Divide by (n−1) = 4: Variance = 74/4 = 18.5
Step 6: Take square root: s = √18.5 = 4.30
💡 Key Insight
The standard deviation is the square root of the variance. We use the square root to return to the original units — if data is in pounds sterling, variance would be in pounds squared (meaningless), but standard deviation is back in pounds. This is why standard deviation is more interpretable and more widely reported than variance.
The Empirical Rule (68-95-99.7 Rule)
For data that follows a normal distribution, standard deviation allows powerful probabilistic statements:
📊 68% of data falls within 1 standard deviation of the mean (μ ± σ)
📊 95% of data falls within 2 standard deviations of the mean (μ ± 2σ)
📊 99.7% of data falls within 3 standard deviations of the mean (μ ± 3σ)
⚠️ Common Error
Students often divide by n rather than (n−1) when calculating a sample standard deviation. The (n−1) denominator — called Bessel’s correction — corrects for the fact that a sample tends to underestimate the population spread. Always use (n−1) for sample data; only use N for a full population census.
Q1. A sample of 5 student test scores is: 72, 85, 90, 68, 75. Calculate the sample standard deviation. [5 marks]
Answer: Mean = (72+85+90+68+75)/5 = 390/5 = 78. Deviations: −6, +7, +12, −10, −3. Squared deviations: 36, 49, 144, 100, 9. Sum = 338. Variance = 338/(5−1) = 338/4 = 84.5. Standard deviation = √84.5 = 9.19. This means scores typically deviate from the mean by about 9.19 marks.
References
1. Moore, D.S., McCabe, G.P. and Craig, B.A. (2021) Introduction to the Practice of Statistics. W.H. Freeman.
2. Triola, M.F. (2022) Elementary Statistics. Pearson.
3. Field, A. (2018) Discovering Statistics Using IBM SPSS Statistics. SAGE.
4. Freedman, D., Pisani, R. and Purves, R. (2007) Statistics. W.W. Norton.
5. Ross, S.M. (2020) Introduction to Probability and Statistics for Engineers and Scientists. Elsevier.