havo wiskunde A vwo wiskunde A
Normal distribution calculator
Quick answer
About 68% lies between μ − σ and μ + σ, about 95% between μ − 2σ and μ + 2σ, and almost 100% between μ − 3σ and μ + 3σ.
μ = 170 and σ = 8: about 68% lies between 162 and 178, about 95% between 154 and 186.
Calculator
Choose what you want to know: the percentage between two bounds, where the bounds of 68% and 95% lie, or how large σ is when you know a percentage. The calculator works with the three rules of thumb and also shows the exact value. Decimals appear with a comma (81,5% rather than 81.5%), as in Dutch schools.
What is the normal distribution?
Many measurements have a bell-shaped distribution: most observations lie close to the mean, and the further you move away from it, the fewer there are. The left and right sides mirror each other. Such a distribution is called a normal distribution (Dutch: normale verdeling).
Two numbers describe it: the mean μ (the center) and the standard deviation σ (how wide the bell is). To find the standard deviation of a list of numbers, use the standard deviation calculator.
The rules of thumb (Dutch: vuistregels) tell you, without a table or formula, what share of the observations lies between two bounds, as long as those bounds are a whole number of σ away from μ. By symmetry the three rules give you each slice of the bell: 68 ÷ 2 = 34% between μ and μ + σ, (95 − 68) ÷ 2 = 13,5% between μ + σ and μ + 2σ, and (100 − 95) ÷ 2 = 2,5% between μ + 2σ and μ + 3σ. Each side of μ holds half, 50%.
Examples
Say the heights of a group of students are normally distributed with μ = 170 cm and σ = 8 cm.
What percentage is between 162 and 186 cm tall?
- Bounds in steps of σ(162 − 170) ÷ 8 = -1, so 162 = μ − σ; (186 − 170) ÷ 8 = 2, so 186 = μ + 2σ
- Add up34 + 34 + 13,5 = 81,5%
About 81,5% of the students are between 162 and 186 cm tall.
Bags of sugar weigh 1000 grams on average, with σ = 10 grams. What percentage weighs more than 1020 grams?
- Bound(1020 − 1000) ÷ 10 = 2, so 1020 = μ + 2σ
- Rule of thumb(100 − 95) ÷ 2 = 2,5
About 2,5% of the bags weigh more than 1020 grams.
The other way round: μ = 1000 grams and 2,5% weighs more than 1020 grams. How large is σ?
- Which bound?2,5% lies above μ + 2σ, so 1020 = μ + 2σ
- Solveσ = (1020 − 1000) ÷ 2 = 10
The standard deviation is 10 grams.
Common mistakes
Wrong
68% lies between μ and μ + σ.
Right
68% lies between μ − σ and μ + σ, so on both sides together. Between μ and μ + σ lies half of it: 34%.
Wrong
5% lies above μ + 2σ.
Right
5% lies outside μ ± 2σ, split over two sides: 2,5% lies above μ + 2σ.
Wrong
With 2,5% above 1020 grams and μ = 1000, saying σ = 20.
Right
1020 is μ + 2σ, so 2σ = 20 and σ = 10.
Wrong
Using the rules of thumb for a bound that is not a whole number of σ from μ, such as 175 with μ = 170 and σ = 8.
Right
Check (bound − μ) ÷ σ first. The rules of thumb only help when it is 1, 2 or 3 (or minus one of those).
To practice with an explanation for every mistake, try normal distribution practice.
The normal distribution on the Dutch exam
The 2027 central exam syllabus for wiskunde A havo spells out the three rules of thumb in domain E (Statistics): about 68% of the observations between μ − σ and μ + σ, about 95% between μ − 2σ and μ + 2σ, and almost 100% between μ − 3σ and μ + 3σ. Subdomain E3 asks you to use those three rules for an (approximately) normal distribution.
In wiskunde A vwo, statistics and probability (domain E) belong to the school exam, not the central exam. In the list of concepts for the central exam, the normal distribution and the three rules of thumb are only ticked for havo wiskunde A.
Next step
- How does this work? The explanation and examples are further up this page.
- Test yourself Normal distribution practice, with an explanation after every answer.
Learning path
- Learn first: Standard deviation calculator Spread: population and sample standard deviation.
Frequently asked questions
What are the rules of thumb for the normal distribution?
About 68% of the observations lie between μ − σ and μ + σ, about 95% between μ − 2σ and μ + 2σ, and almost 100% between μ − 3σ and μ + 3σ. μ is the mean and σ the standard deviation.
What percentage lies above μ + 2σ?
About 2,5%. Outside μ ± 2σ lies 100 − 95 = 5%, and by symmetry that is 2,5% on each side.
Why do the rules of thumb say “about”?
The rules of thumb are rounded numbers. Computed exactly with the normal distribution, 68,27% lies between μ − σ and μ + σ. When a question mentions the rules of thumb, you work with 68%, 95% and almost 100%.