∑Richard Hummel Math

Normal Distribution

The symmetric bell-shaped distribution that arises naturally from sums of many small random effects.

Practice exercises →
The 68–95–99.7 rule
−3σ−2σ−σμ+σ+2σ+3σ68%95%99.7%
Definition

The normal distribution is a bell-shaped, symmetric curve that describes how many real-world quantities are distributed. It is centred on the mean μ\mu and its spread is controlled by the standard deviation σ\sigma.

Key features:

  • Symmetric around the mean
  • The mean, median, and mode are all equal
  • Most values cluster near the centre; extreme values become increasingly rare
Definition

The 68–95–99.7 rule (empirical rule):

  • About 68% of values fall within 1σ1\sigma of the mean
  • About 95% fall within 2σ2\sigma
  • About 99.7% fall within 3σ3\sigma
Heights of adults

Adult heights in a population are approximately normal with mean μ=170 cm\mu = 170\,\text{cm} and σ=10 cm\sigma = 10\,\text{cm}.

  • About 68% of people are between 160 cm and 180 cm
  • About 95% are between 150 cm and 190 cm
  • Virtually everyone (99.7%) is between 140 cm and 200 cm
Try it

IQ scores are normally distributed with mean 100 and standard deviation 15. Approximately what percentage of people have IQ between 70 and 130?

Solution

70 and 130 are each 3015=2\frac{30}{15} = 2 standard deviations from the mean. By the empirical rule, approximately 95% of people fall in this range.

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