A symmetric bell curve: A normal variable X ~ N(μ, σ²) has a bell-shaped graph symmetric about its mean μ. The standard deviation σ sets the width. Probabilities are areas under the curve, and the total area is 1.
IB-style question — read the parameters
The masses of apples are modelled by X ~ N(150, 20²) grams. State the mean and standard deviation, and the value the distribution is symmetric about.
Step by step
- N(μ, σ²): first is the mean, second is the variance.
- Symmetric about the mean.
Final answer
Mean 150 g, standard deviation 20 g, symmetric about 150 g.
Second number is the variance: N(150, 20²) means σ = 20 (variance 400). The GDC needs σ, so take the square root of the variance if you're given it.
Area between a lower and upper bound: On Paper 2 use normalcdf(lower, upper, μ, σ). For P(X < a) use a very small lower bound; for P(X > a) use a very large upper bound; for P(a < X < b) use both.
IB-style question — P(X < a)
X ~ N(50, 8²). Find P(X < 60) and P(X > 60).
Step by step
- P(X < 60) = normalcdf(−∞, 60, 50, 8).
- P(X > 60) is the complement.
Final answer
P(X < 60) ≈ 0.894; P(X > 60) ≈ 0.106.
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Standard areas you can use by hand: Without a calculator, use symmetry (P(X < μ) = 0.5) and the empirical rule: about 68% of data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.
IB-style question — empirical rule
Heights are X ~ N(170, 10²) cm. Use the empirical rule to estimate P(160 < X < 180).
Step by step
- 160 and 180 are μ − σ and μ + σ.
- Within 1σ → empirical rule.
Final answer
About 0.68 (68%).
Split the leftover symmetrically: Outside 1σ is 1 − 0.68 = 0.32, split into two equal tails of 0.16 each — handy for P(X > μ + σ).
Probability × how many: To find an expected number meeting a condition, find the probability with normalcdf, then multiply by the total number — exactly like n × P.
IB-style question — how many
Bolt lengths are X ~ N(50, 2²) mm. A bolt is rejected if it is longer than 53 mm. In a batch of 400 bolts, find the expected number rejected.
Step by step
- Probability one bolt is too long.
- Expected number = probability × 400.
Final answer
About 27 bolts are expected to be rejected.
Two steps, both shown: Show the probability and the × total — each usually earns a mark.