Normal Distribution Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Normal Distribution.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The normal distribution (also called the Gaussian distribution or bell curve) is a continuous probability distribution that is symmetric about its mean, with data tapering off equally on both sides following a precise mathematical rule.

The normal distribution describes data that clusters symmetrically around the mean with a characteristic bell shape — most values are near the mean, and extreme values become rapidly less likely.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: 68-95-99.7 rule: 68\% within 1 SD, 95\% within 2 SD, 99.7\% within 3 SD.

Common stuck point: Not everything is normal—income and city sizes follow different distributions.

Sense of Study hint: Sketch a bell curve, mark the mean at center, then mark 1, 2, and 3 SDs on each side. Use the 68-95-99.7 rule to estimate areas.

Worked Examples

Example 1

medium
Scores on a test are normally distributed with mean \mu = 75 and standard deviation \sigma = 10. What percentage of students scored between 65 and 85?

Solution

  1. 1
    Identify the mean \mu = 75 and standard deviation \sigma = 10. Check whether 65 and 85 are within one standard deviation.
  2. 2
    Verify: 65 = 75 - 10 = \mu - \sigma and 85 = 75 + 10 = \mu + \sigma, so the interval [65, 85] is exactly \mu \pm \sigma.
  3. 3
    By the empirical rule (68-95-99.7 rule), approximately 68\% of data in a normal distribution falls within one standard deviation of the mean.

Answer

\approx 68\%
The empirical rule provides quick approximations for normal distributions: about 68\% within 1\sigma, 95\% within 2\sigma, and 99.7\% within 3\sigma of the mean.

Example 2

medium
Heights of adult women are normally distributed with \mu = 164 cm and \sigma = 6 cm. What percentage of women are taller than 176 cm?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

hard
A machine fills bottles with a mean of 500 mL and standard deviation 4 mL (normally distributed). What percentage of bottles contain between 492 mL and 508 mL?

Example 2

medium
A standardized exam has scores that are approximately normal with mean 500 and standard deviation 100. About what percent of students score between 400 and 700?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

meanstandard deviation