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.
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.
[Diagram: math-normal-curve] - Available in full study mode
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.