Symmetric bell; mean = median = mode: The normal curve is a symmetric bell centred on the mean.
Because it's symmetric, the mean = median = mode, the total area is 1, and half the area (0.5) lies on each side of the mean.
IB-style question — use the properties
X ~ N(40, σ²).
Write down P(X < 40) and P(X > 40).
Step by step
- Symmetric about the mean 40.
- The other half.
Final answer
P(X < 40) = P(X > 40) = 0.5.
Mean splits the area in half: Anything asked at the mean is 0.5 — no calculator needed.
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Draw the bell, mark μ, shade the region: To show a probability, sketch the bell with the mean in the middle, mark the boundary value(s), and shade the region whose area you want.
The shaded area is the probability.
The shaded area is the probability — switch between a left tail P(X<a), a right tail P(X>a), and the central 68%.
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IB-style question — describe the shading
X ~ N(60, 5²).
Describe how to represent P(X > 68) on a sketch.
Step by step
- Mark the mean and the boundary.
- Shade to the right of 68.
Final answer
Draw the bell centred at 60, mark 68 to the right of centre, and shade the tail beyond 68.
A sketch checks your answer: If your probability is large but the shaded tail is tiny (or vice versa), you've made an error.
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Mean moves the centre; σ changes the width: A different mean (same σ) shifts the curve left/right without changing its shape.
A larger σ makes the curve wider and flatter (more spread); a smaller σ makes it taller and narrower.
Change the mean μ
- shifts the curve left/right
- same shape and width
- centre moves to the new mean
Change σ
- larger σ → wider, flatter
- smaller σ → taller, narrower
- same centre
IB-style question — compare
Two groups' marks are normal with the same mean but Group A has σ = 4 and Group B has σ = 9.
Which curve is taller and narrower, and what does that mean?
Step by step
- Smaller σ → taller, narrower.
- Interpret.
Final answer
Group A's curve is taller and narrower (σ = 4), meaning its marks are more clustered/consistent.