4 · IB Maths AA
Standard deviation and outliers
Drag the data points on the top number line and watch the mean, standard deviation, quartiles and 1.5 × IQR fences respond. The lower line shows the transformed data y = ax + b.
| Statistic | x (original) | y = ax + b |
|---|
Selected point: –
x̄ = Σx ÷ n
σ = √( Σ(x − x̄)² ÷ n )
IQR = Q₃ − Q₁
Outlier if x < Q₁ − 1.5 × IQR or x > Q₃ + 1.5 × IQR
y = ax + b ⇒ ȳ = ax̄ + b, σᵧ = |a|σₓ, IQRᵧ = |a|·IQRₓ
σ = √( Σ(x − x̄)² ÷ n )
IQR = Q₃ − Q₁
Outlier if x < Q₁ − 1.5 × IQR or x > Q₃ + 1.5 × IQR
y = ax + b ⇒ ȳ = ax̄ + b, σᵧ = |a|σₓ, IQRᵧ = |a|·IQRₓ
Try this
- Choose "Roughly symmetric" and drag the largest value to 20. Compare how much the mean and σ change with how much the median and IQR change.
- Keep a = 1 and set b = 10. Which statistics move by 10, and which do not change at all?
- Set a = 2 with "With an outlier". Check that σ and IQR both double, and decide whether the outlier is still an outlier.
Exam tip: for transformed data y = ax + b, write ȳ = ax̄ + b and σᵧ = |a|σₓ immediately — the added constant b shifts the centre but never changes the spread.