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.
The shaded area under the normal curve IS the probability — a left tail, a right tail, or a central region.
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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.
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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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IB-style question — find σ from the IQR
A normally distributed variable has interquartile range 13.49.
Find its standard deviation σ.
Step by step
- The quartiles sit 0.6745 standard deviations either side of the mean (invNorm(0.75) = 0.6745), so the IQR spans 2 × 0.6745 = 1.349 standard deviations.
- Solve for σ.
Final answer
σ = 10.
Q1 and Q3 lie 0.6745σ either side of the mean, so the IQR = 1.349σ — divide the IQR by 1.349 to recover σ.
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IB-style question — conditional on a normal region
X ~ N(20, 4²). Find P(X > 24 | X > 18).
Step by step
- Conditional: keep only X > 18, take the X > 24 slice.
- Each probability from normalcdf on the GDC.
Final answer
≈ 0.230.
IB-style question — find μ and σ from two probabilities
X ~ N(μ, σ²), with P(X < 50) = 0.2 and P(X < 70) = 0.9.
Find μ and σ.
Step by step
- Standardise each probability using invNorm to get the z-values, then write two equations.
- Subtract to eliminate μ.
- Back-substitute for μ.
Final answer
μ ≈ 57.9, σ ≈ 9.42.
Two given probabilities give two z-equations; subtracting eliminates μ to find σ, then back-substitute for μ.
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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.