Normal Distribution (AI SL)
The normal distribution \(N(\mu,\sigma^2)\) models continuous data that clusters symmetrically around a mean - heights, race times, exam marks. Every probability question on this bell-shaped curve is answered on your GDC, either forwards (given a value, find a probability) or backwards (given a probability, find the value). It's part of the broader Distributions topic.
15 questions on this sub-topic.
Two GDC techniques
Covered under IB syllabus reference SL4.9: the normal distribution and curve, including that approximately 68% of data lies within \(\mu\pm\sigma\), 95% within \(\mu\pm2\sigma\), and 99.7% within \(\mu\pm3\sigma\). Both normal probability calculations and inverse normal calculations are done using technology, not by hand.
Normal probability
Find \(P(X
Inverse normal
Given a probability (area to the left), find the corresponding value of \(X\) using your GDC's inverse normal function - \(\mu\) and \(\sigma\) are always given.
Not in the formula booklet - GDC requiredNeed the full syllabus wording and formula-booklet reference table? See Distributions.
Worked examples
Times to run a race are \(N(45, 6^2)\) seconds.
Find the probability a runner finishes in under 40 seconds (3 significant figures).
Worked solution
Times \(X\sim N(45,6^2).\) ‘Under 40 seconds’ means the left-hand tail \(P(X<40).\) M1 A1
\(40\) is \(\tfrac{40-45}{6}\approx -0.83\) standard deviations below the mean, so expect a probability around \(0.2.\)
Exam marks are \(N(58, 12^2)\). The top 15% receive a distinction.
Find the minimum mark for a distinction.
Worked solution
Top 15% means \(P(X>k)=0.15,\) so \(P(X<k)=0.85.\) M1
\(k=\text{invNorm}(0.85,58,12)\approx 70.4.\) A1 Since more than the cutoff must be excluded, round up: the minimum mark for a distinction is \(71.\) A1
Common mistakes
- Rounding invNorm the wrong way. When a boundary needs a whole number and the question says "at least" or "more than" a cutoff, check carefully whether to round up or down - rounding the wrong direction moves people across the cutoff.
- Muddling "less than" and "more than" bounds. normalcdf needs a lower and upper bound in that order - for "less than \(a\)" the lower bound is a very large negative number, and for "more than \(a\)" the upper bound is a very large positive number. Swapping them silently gives the wrong tail.
- Using the normal distribution for a discrete count. A count of successes from a fixed number of trials (e.g. number of heads in 20 flips) is binomial, not normal - check whether the data is discrete or continuous before choosing a model.
Ready to practise properly?
15 normal-distribution questions, marked instantly like the real exam.
Quick answers
How do I find a normal probability on my GDC?
Use the normal cdf function with the lower bound, upper bound, mean, and standard deviation. For "less than" use a very large negative lower bound; for "greater than" use a very large positive upper bound. See the GDC guidance on the full Distributions page.
What does the inverse normal function do?
Given an area (probability) to the left, inverse normal returns the corresponding value of \(X\), using the mean and standard deviation you supply.