Normal Distribution Examples in Statistics

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 Statistics.

Concept Recap

A symmetric, bell-shaped probability distribution where most data clusters around the mean, with probabilities decreasing symmetrically toward the tails.

Heights, test scores, measurement errors - many real phenomena cluster around an average with decreasing frequency toward extremes. The bell curve captures this pattern: most values are 'average,' few are extreme.

Read the full concept explanation โ†’

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: The normal distribution is bell-shaped and symmetric about the mean. About 68% of data falls within one standard deviation, 95% within two, and 99.7% within three.

Common stuck point: Students assume all real data is normally distributed. Many datasets โ€” income, reaction times, test scores โ€” are skewed and require different methods.

Worked Examples

Example 1

medium
IQ scores follow a normal distribution with mean \mu = 100 and standard deviation \sigma = 15. Using the 68-95-99.7 rule, what percentage of people have IQs between 85 and 115?

Solution

  1. 1
    Step 1: 85 is one standard deviation below the mean: 100 - 15 = 85.
  2. 2
    Step 2: 115 is one standard deviation above the mean: 100 + 15 = 115.
  3. 3
    Step 3: By the 68-95-99.7 rule, approximately 68% of data falls within one standard deviation of the mean.

Answer

Approximately 68%.
The empirical rule (68-95-99.7) gives quick estimates for normal distributions: about 68% within 1ฯƒ, 95% within 2ฯƒ, and 99.7% within 3ฯƒ of the mean.

Example 2

hard
Birth weights are normally distributed with \mu = 3.5 kg and \sigma = 0.5 kg. What percentage of babies weigh between 2.5 and 4.5 kg?

Practice Problems

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

Example 1

medium
A factory produces bolts with mean length 10 cm and standard deviation 0.2 cm (normally distributed). What percentage of bolts are longer than 10.4 cm?

Example 2

medium
A normal distribution has mean \mu = 50 and standard deviation \sigma = 4. Using the 68-95-99.7 rule, what percentage of values lie between 42 and 58?
โ† Back to Normal Distribution Practice Mode

Background Knowledge

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

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