IB Maths AI HLNormal DistributionPaper 1 & 2~7 min read
Calculations with Normal Distribution
With X ∼ N(μ, σ2), every probability is an area under the bell curve, found on your GDC. Use the normal CD (cumulative) function for any range — never the normal PD, which gives density, not probability. For one-sided tails, plug in a “very big” bound. And to go backwards from a probability to a value of x, use the inverse normal. A quick sketch of the area each time keeps you out of trouble.
📘 What you need to know
P(X = x) = 0 for a continuous variable, so strict and weak inequalities are equal.
Use normal CD (NCD / Normal Cdf) for P(a < X < b) — ignore normal PD on this course.
CD inputs: lower bound, upper bound, μ, and σ (the standard deviation — root the variance if needed).
One-sided tails: for P(X > a) use a huge upper bound (e.g. 1099); for P(X < b) use a huge negative lower bound.
Useful identities: P(X < μ) = 0.5; P(X > a) = 1 − P(X < a); P(a < X < b) = P(X < b) − P(X < a).
Inverse normal (InvN): given an area to the left, returns the value a with P(X < a) = p.
Always sketch and sense-check: small left-area → value below the mean, and vice versa.
Probabilities with the normal CD
The area under the curve between a and b is P(a < X < b). Because P(X = x) = 0, you never worry about < versus ≤. Find the area with the normal CD function.
Range probability = area
P(a < X < b) = P(a ≤ X ≤ b) = normal CD(a, b, μ, σ)
use the normal CD on your GDC — NOT the normal PD
🧭 Recipe — P(a < X < b) on the GDC
Identifyμ and σ (square-root the variance if you’re given σ2).
Sketch the curve and shade the area you want.
Open the normal CD function and enter lower, upper, μ, σ (mind the input order).
Read off the probability and round (usually 3 sf).
🧠 Memory aid — “CD not PD”
The normal PD (Pdf) gives the height of the curve at a point — useless for probability, and zero-width anyway. Always reach for the normal CD (Cdf), which gives the area. PD = point/density, CD = cumulative/area.
WE 1
A single value
Y ∼ N(20, 52). Find P(Y = 20).
single value of a continuous variablethe area of a single line is zero.P(Y = 20) = 0always zero for any exact value of a normal variable.
WE 2
A range probability
Y ∼ N(20, 52). Find P(18 ≤ Y < 27).
identify μ and σμ = 20, σ = 5 (already given as 5²)use normal CD on GDClower = 18, upper = 27 → P = 0.574665…P(18 ≤ Y < 27) ≈ 0.575 (3sf)≤ and < make no difference for a normal variable.
One-sided tails & identities
For a one-sided probability you still use the normal CD, but one bound is “infinity”. The trick is to enter a number far enough into the tail that the missing area is negligible.
P(X > a) — upper tail
huge upper
lower = a, upper = a very big number like 1099 (or 99999…).
P(X < b) — lower tail
huge negative
lower = a very big negative like −1099, upper = b.
Useful identities
P(X < μ) = P(X > μ) = 0.5
P(X > a) = 1 − P(X < a) P(a < X < b) = P(X < b) − P(X < a)
handy when μ/σ are unknown or you only have a diagram
An upper tail: P(Y > 29) on N(20, 5²)
For an upper tail set lower = 29, upper = a very big number; here the area is ≈ 0.0359.
WE 3
An upper-tail probability
Y ∼ N(20, 52). Find P(Y > 29).
no upper bound → use a big numberlower = 29, upper = 99999use normal CD on GDCP(Y > 29) = 0.035930…P(Y > 29) ≈ 0.0359 (3sf)29 is well above the mean of 20, so a small tail probability makes sense.
The inverse normal
The inverse normal (InvN) goes the other way: you give it an area to the left and it returns the value of x. To use it, every probability must first be written as a left tail P(X < a).
🧭 Recipe — inverse normal
Rewrite as a left tail: if you’re given P(X > a) = p, use P(X < a) = 1 − p.
Open InvN and enter the area (left), then μ, then σ.
Read off the value of a.
Sense-check: left-area < 0.5 → a below μ; left-area > 0.5 → a above μ.
🤔 Why turn P(X > a) into a left tail?
The basic inverse normal expects the area to the left of the value. If a question gives a right-tail probability, subtract it from 1 first: P(X < a) = 1 − P(X > a). (Some calculators have a “tail: right” option that skips this step.) Then the calculator can find the matching a.
WE 4
Inverse normal from a right tail
W ∼ N(50, 36). Find the value of w such that P(W > w) = 0.175.
identify μ and σμ = 50, σ² = 36 → σ = 6rewrite as a left tailP(W < w) = 1 − 0.175 = 0.825use InvN with area 0.825w = 55.6075…w ≈ 55.6 (3sf)area 0.825 > 0.5, so w is above the mean of 50 — checks out.
WE 5
Inverse normal from a left tail
W ∼ N(50, 36). Find the value of w such that P(W < w) = 0.30.
already a left tailarea = 0.30, μ = 50, σ = 6use InvN directlyw = 46.853…w ≈ 46.9 (3sf)area 0.30 < 0.5, so w is below the mean — makes sense.
💡 Top tips
Use normal CD, never normal PD — CD gives area (probability), PD gives density.
Square-root the variance before entering σ into the GDC.
Sketch the area every time so you know what you’re calculating.
One-sided tails: use ±1099 (or lots of 9s) for the “infinite” bound.
Inverse normal needs a left area — convert right tails with 1 − p first.
Sense-check the answer: left-area below 0.5 means a value below the mean.
⚠ Common mistakes
Entering the variance as σ — the GDC wants the standard deviation.
Using normal PD for a probability — it returns density, not area.
Wrong input order — some calculators ask for σ before μ; check carefully.
Feeding a right-tail area straight into InvN without converting to 1 − p.
Too-small “infinity” for a tail — use a number well beyond 4σ from the mean.
Not sense-checking — an answer on the wrong side of the mean signals a slip.
That completes the Normal Distribution unit! You can now describe a normal model, read off its mean and standard deviation, find probabilities with the normal CD, handle one-sided tails, and run the inverse normal to recover a value from a probability. These GDC skills carry straight into hypothesis testing and confidence intervals, where normal probabilities underpin critical values and test statistics.
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