Practice Normal distribution
A normal distribution is a symmetric bell-shaped model determined by its mean and standard deviation.
Probabilities correspond to areas under the curve.
Exercises with solutions
The normal distribution is symmetric about the expected value μ and its shape depends only on μ and the standard deviation σ. Probabilities are found using the standardised value z = (x − μ) ÷ σ. For large n the binomial distribution can be approximated by the normal distribution.
Exercise 1
What is the probability that a value lies between μ − σ and μ + σ?
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about 68 %
By the sigma rule about 68 % of values lie within μ ± 1σ.
Exercise 2
What percentage of values lie outside μ ± 2σ?
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about 5 %
About 95 % lie within μ ± 2σ, so roughly 5 % lie outside.
Exercise 3
What do you need the standardisation z = (x − μ) ÷ σ for?
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to use the same table for every normal distribution
With z you convert any normal distribution to the standard normal distribution.
Exercise 4
When may you approximate a binomial distribution by the normal distribution?
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for large n
As a rule of thumb the approximation is usable when σ = √(n·p·(1−p)) is greater than 3.
Common mistake
Treating the sigma rules as exact values. They are approximations; for exact values you need the table. And overlooking that 1σ, 2σ and 3σ give different percentages.
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