Inverse Normal and Parameters (AI HL)
A normal probability question usually gives you a value and asks for a probability. An inverse normal question flips that around: you're given the probability (an area under the curve) and asked to find the boundary value, or even an unknown mean or standard deviation, that produces it. These questions are almost always solved on the GDC, but setting up the right equation first is what actually earns the marks. It's part of the broader Probability & Distributions topic.
25 questions on this sub-topic.
Standardising and going backwards
Covered under IB syllabus reference SL4.9. Neither relationship below is in the formula booklet - you're expected to know the standardising equation and to drive the GDC's inverse normal routine confidently.
Standardising a value
\(z = \dfrac{x-\mu}{\sigma}\)
Converts any normal variable \(X\sim N(\mu,\sigma^2)\) to the standard normal \(z\)-score. Useful when a parameter (\(\mu\) or \(\sigma\)) is the unknown, since it turns the problem into an equation.
Not in the formula bookletInverse normal
Given \(P(X Always enter the area to the left of the unknown value - if you're told "the top \(k\%\)", convert it to \(1-\tfrac{k}{100}\) first.
Need the full syllabus wording and formula-booklet reference table? See Probability & Distributions. GDC keystrokes for these problems are in the same page's GDC section.
Worked examples
Lifetimes of a battery are normally distributed with mean 80 hours and standard deviation 8 hours.
(a) Find the lifetime exceeded by 90% of batteries.
(b) The manufacturer guarantees a minimum lifetime so that only 5% fail early. Find that minimum lifetime.
Worked solution
(a) \(P(X > L) = 0.90 \Rightarrow P(X < L) = 0.10\) M1
\(L \approx 69.7\) h. A1
(b) \(P(X < m) = 0.05\) M1
\(m \approx 66.8\) h. A1
\(m \approx 66.8\) h.
IQ scores are normal with mean 100 and standard deviation 15.
(a) Find the upper quartile.
(b) Find the interquartile range.
Worked solution
(a) \(P(X < Q_3) = 0.75\) M1
\(Q_3 \approx 110.1.\) A1
(b) By symmetry M1
\(Q_1 \approx 89.9\) A1
IQR \(\approx 20.2.\)
Common mistakes
- Forgetting inverse normal needs the area to the LEFT. For "the top 10%", the area to enter is \(0.90\), not \(0.10\) - misreading this flips the answer to the wrong tail entirely.
- Not identifying which quantity is actually unknown. Sometimes it's \(x\) you're solving for, sometimes it's \(\mu\) or \(\sigma\) - each needs the standardising equation rearranged differently, and treating every question as "find \(x\)" leads to the wrong setup.
- Rounding a \(z\)-value or intermediate probability too early. Carrying a rounded \(z\)-score into a further calculation for \(\mu\) or \(\sigma\) can shift the final answer outside the accuracy the mark scheme accepts - keep full GDC precision until the last line.
Ready to practise properly?
25 inverse-normal questions, marked instantly like the real exam.
Quick answers
What does "inverse normal" mean?
A normal probability calculation works forwards: given \(x\), find \(P(X
Which area do I enter for "the top 10%"?
Inverse normal functions on the GDC always take the area to the left of the unknown value. For "the top 10%", the boundary has 90% of the distribution below it, so you enter \(0.90\), not \(0.10\).