How spread out the data is, about the mean: Standard deviation (σ) measures how far values typically lie from the mean.
A small σ means the data is clustered; a large σ means it is spread out.
Variance is just σ² (the standard deviation squared).
Standard deviation measures spread: ~68% of a normal data set lies within one standard deviation of the mean.
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IB-style question — compare spreads
Two classes have the same mean test score.
Class A has standard deviation 4; Class B has standard deviation 11.
Which class is more consistent, and why?
Step by step
- Smaller σ ⇒ values closer to the mean.
- Interpret.
Final answer
Class A is more consistent — its smaller standard deviation means scores are closer to the mean.
Variance = σ²: If a question asks for the variance, square the standard deviation (and vice versa: σ = √variance).
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Enter the data, read σx from 1-Var Stats: On Paper 2 you find the standard deviation with 1-Var Stats: enter the values (and frequencies if given), run it, and read σx (the population standard deviation the IB syllabus uses).
IB-style question — mean and σ
Find the mean and standard deviation of 2, 4, 4, 6, 9.
Step by step
- Mean = sum ÷ n.
- 1-Var Stats gives σx (population SD).
Final answer
Mean = 5, standard deviation ≈ 2.37.
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Adding shifts the mean; multiplying scales both: Add c to every value: the mean increases by c, the standard deviation is unchanged (the spread doesn't move).
Multiply by k: the mean and standard deviation are both multiplied by |k|.
IB-style question — transform the data
A data set has mean 20 and standard deviation 4.
Every value is increased by 5, then the result is doubled.
Find the new mean and standard deviation.
Step by step
- Add 5: mean + 5, SD unchanged.
- Double: multiply both by 2.
Final answer
New mean = 50, new standard deviation = 8.
Adding does NOT change spread: Shifting every value by the same amount slides the whole data set — distances from the mean stay the same, so σ is unchanged.