Formula sheet
Standard deviation (ungrouped)
Both forms σ = √(Σ(x−x̄)²/N) = √(Σx²/N − x̄²) are given on the SPM formula sheet. You must know that σ is just the square root of the variance, so do not forget the final √, and remember σ carries the same unit as the original data.
What the symbols mean
- σ the standard deviation, in the same unit as the data
- x each individual data value
- x̄ the mean of the data
- N the number of data values
- Σ(x−x̄)² the sum of the squared deviations from the mean
- Σx² the sum of the squares of the data values
Given in the exam, or memorise?
Both forms σ = √(Σ(x−x̄)²/N) = √(Σx²/N − x̄²) are given on the SPM formula sheet. You must know that σ is just the square root of the variance, so do not forget the final √, and remember σ carries the same unit as the original data.
Why it works
The standard deviation undoes the squaring in the variance, so the spread is measured back in the data's own unit.
- Compute the variance σ² using either form.
- The variance is in squared units, which is hard to interpret directly.
- Take the square root of the variance to return to the original unit: σ = √(σ²).
- The result σ is a typical distance of a value from the mean.
Worked example 1
Find the standard deviation of 5, 7, 3, 9, 6.
- Find the mean: x̄ = (5 + 7 + 3 + 9 + 6) / 5 = 30 / 5 = 6.
- Sum of squares: Σx² = 25 + 49 + 9 + 81 + 36 = 200.
- Variance: σ² = Σx²/N − x̄² = 200/5 − 6² = 40 − 36 = 4.
- Square root: σ = √(Σx²/N − x̄²) = √4.
Worked example 2
Find the standard deviation of 10, 12, 14, 16, 18, giving the answer to 2 decimal places.
- Find the mean: x̄ = (10 + 12 + 14 + 16 + 18) / 5 = 70 / 5 = 14.
- Deviations x − x̄: −4, −2, 0, 2, 4; squares 16, 4, 0, 4, 16, so Σ(x−x̄)² = 40.
- Variance: σ² = Σ(x−x̄)²/N = 40 / 5 = 8.
- Square root: σ = √8 = 2.8284...
Where students go wrong
- Reporting the variance as the standard deviation, forgetting the final square root is the single most common slip.
- Taking the square root of Σ(x−x̄)² alone, without first dividing by N.
- Rounding the variance too early, then square-rooting a rounded number; keep full accuracy until the last step.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
What is the difference between variance and standard deviation?
Standard deviation is the square root of the variance: σ = √(σ²). The variance is in squared units and is a stepping stone, while the standard deviation is in the same unit as the data, so it is easier to interpret as a typical spread.
Compute the variance first, then take the root.
Why is the standard deviation often not a whole number?
Because it comes from a square root, and most variances are not perfect squares. A variance of 8 gives σ = √8 ≈ 2.83, which is irrational.
Keep the exact value in your calculator and round only at the end, usually to two or four significant figures as the question requires.
What does a larger standard deviation tell me?
A larger standard deviation means the data is more spread out from the mean; the values are less consistent. A smaller one means the data clusters tightly around the mean.
When comparing two data sets with a similar mean, the set with the smaller standard deviation is the more uniform or reliable one.