Normal Distribution (AI HL)
The normal distribution is the familiar symmetric bell curve used to model continuous data that clusters around a mean - heights, test scores, measurement error. This page covers its key properties and how to find probabilities from it on the GDC, with worked examples and the mistakes that come up most. It's part of the broader Probability & Distributions topic.
11 questions on this sub-topic.
Properties and the spread of data
Covered under IB syllabus reference SL4.9. Probabilities from the normal distribution are always found using technology - there's no formula to substitute into, but knowing the shape helps you sanity-check the GDC's answer.
Standardising a value
\(z = \dfrac{x-\mu}{\sigma}\)
Converts \(X\sim N(\mu,\sigma^2)\) to the standard normal \(z\)-score - useful for reasoning about where a value sits, even though the GDC works directly with \(\mu\) and \(\sigma\).
Not in the formula bookletThe 68-95-99.7 rule
About \(68\%\) of data lies within \(1\sigma\) of \(\mu\), \(95\%\) within \(2\sigma\), and \(99.7\%\) within \(3\sigma\).
A quick way to sanity-check an answer without a calculator - if a value is close to two standard deviations away, the probability should be close to 5% in the tail.
Not in the formula booklet - prior knowledgeNeed 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
A variable is \(N(50, 6^{2}).\) State the probability that a value is greater than the mean of 50.
Worked solution
A normal distribution is symmetric about its mean, so exactly half the area lies on each side. R1
\(P(X>50) = 0.5.\) A1 (No calculation or GDC needed - 50 is the mean.)
Scores on a standardized test are normally distributed with mean \(500\) and standard deviation \(100.\) Find the probability that a randomly selected test-taker scores between \(450\) and \(650.\)
Worked solution
M1
\(P(450<X<650)=0.625.\) A1
Common mistakes
- Treating the normal distribution as discrete. For a continuous variable, \(P(X=x)=0\) exactly - "exactly 50" has no meaningful probability, so questions always ask for probabilities over a range or a single boundary, never a single point.
- Not using the complement when only one bound is given. If asked for \(P(X>a)\), remember the total area under the curve is 1, so \(P(X>a) = 1-P(X
- Skipping a quick sketch or symmetry check. A rough sketch of where the value sits relative to the mean catches an inverted probability (e.g. reading off "below" when the question asks "above") before it costs the final answer mark.
Ready to practise properly?
11 normal-distribution questions, marked instantly like the real exam.
Quick answers
What are the key properties of the normal distribution?
It is symmetric about its mean, with the mean, median and mode all equal. Approximately 68% of data lies within one standard deviation of the mean, 95% within two, and 99.7% within three.
How do I find a normal probability on the GDC?
Use the calculator's normal cumulative distribution function with the mean and standard deviation, plus lower and upper bounds - a very large negative number (such as \(-1\times10^{99}\)) stands in for negative infinity when only an upper bound is given.