Statistics Quest

Track 5 · Probability Distributions · Lesson 6 of 10

The normal distribution

Describe the normal distribution through its mean and standard deviation, apply the 68–95–99.7 rule, and calculate normal probabilities as areas.

30 minIntermediate8 graded exercisesMembers

A bottling line is set to fill 500 ml, but every bottle differs slightly. How many will be under-filled by more than 5 ml? Questions like this — about measurements that cluster symmetrically around a target — are answered with the normal distribution, the most used model in statistics.

The normal distribution is a continuous distribution with a symmetric, bell-shaped density. It is fully described by two parameters:

  • the mean μ\mu, which sets the location — the centre of the bell, where mean, median and mode coincide;
  • the standard deviation σ>0\sigma > 0, which sets the spread — the distance from the centre to the points where the curve changes from bending down to bending up.

We write X∼N(μ,σ2)X \sim N(\mu, \sigma^2). Careful: by convention the second number in that notation is the variance σ2\sigma^2. N(500,16)N(500, 16) means σ=4\sigma = 4, not 16.

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It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:

  • Describe a normal distribution by μ (location) and σ (spread), and read N(μ, σ²) correctly
  • Apply the 68–95–99.7 rule and know its exact values
  • Calculate lower-tail, upper-tail and between probabilities as areas under the normal curve
  • Fit a normal model to data and judge whether it is a sensible description
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