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

The normal distribution (bell curve) is a symmetric, bell-shaped probability distribution where most data clusters around the mean, with probabilities decreasing symmetrically toward the tails. It is defined by two parameters: the mean and the standard deviation.

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.

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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: Normal Distribution asks how a value or feature behaves inside the full distribution.

Common stuck point: Students often know a procedure related to normal distribution but skip the recognition step: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary?

Worked Examples

Example 1

medium
SAT scores have μ=1000\mu = 1000, σ=200\sigma = 200. A score of 14001400 is at what percentile, roughly?

Answer

≈97.5th percentile\approx 97.5\text{th percentile}

First step

1
z=(1400−1000)/200=2z = (1400 - 1000)/200 = 2.

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Example 2

hard
For μ=50\mu = 50, σ=10\sigma = 10, find the probability P(40<X<60)P(40 < X < 60).

Example 3

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

Example 4

hard
Birth weights are normally distributed with μ=3.5\mu = 3.5 kg and σ=0.5\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

easy
What is the shape of a normal distribution's graph?

Example 2

easy
For a normal distribution, the mean, median, and mode are all located where?

Example 3

easy
Which two parameters fully define a normal distribution?

Example 4

easy
In a normal distribution, what percent of data lies below the mean?

Example 5

easy
Approximately what percent of data lies within ONE standard deviation of the mean in a normal distribution?

Example 6

easy
Are the tails of a normal distribution ever exactly reaching zero?

Example 7

easy
Is every bell-shaped distribution exactly normal?

Example 8

easy
A normal distribution has mean 100100. The value 100100 is at what percentile?

Example 9

medium
A normal distribution has μ=50\mu=50, σ=5\sigma=5. Between which two values does about 68%68\% of data lie?

Example 10

medium
A normal distribution has μ=50\mu=50, σ=5\sigma=5. About 95%95\% of data lies between which values?

Example 11

medium
A normal distribution has μ=70\mu=70, σ=4\sigma=4. What value is 22 standard deviations above the mean?

Example 12

medium
In a normal distribution, what percent of data lies ABOVE the mean plus one standard deviation? (Use 68-95-99.7.)

Example 13

medium
Two normal curves share μ=0\mu=0 but curve A has σ=1\sigma=1 and curve B has σ=3\sigma=3. Which is taller and narrower?

Example 14

medium
Test scores are normal with μ=500\mu=500, σ=100\sigma=100. Roughly what percent score above 600600?

Example 15

medium
Why can a normal distribution model heights but not the number of children per family well?

Example 16

medium
A normal distribution has μ=20\mu=20, σ=2\sigma=2. The value 2424 is how many standard deviations above the mean?

Example 17

medium
A normal distribution has μ=30\mu=30, σ=5\sigma=5. The value 2020 is how many standard deviations below the mean?

Example 18

challenge
Heights are normal with μ=170\mu=170 cm, σ=10\sigma=10 cm. Estimate the percent of people taller than 190190 cm using the empirical rule.

Example 19

challenge
Scores are normal with μ=60\mu=60, σ=8\sigma=8. Estimate the percent of scores between 5252 and 7676 using the empirical rule.

Example 20

challenge
Two normal distributions: A has μ=100,σ=15\mu=100,\sigma=15; B has μ=100,σ=30\mu=100,\sigma=30. A value of 130130 is more 'unusual' in which distribution, and why?

Example 21

easy
For a normal distribution, what percent of data lies within ±2\pm 2 SDs of the mean?

Example 22

easy
What percent of data lies within ±3\pm 3 SDs of the mean for a normal distribution?

Example 23

easy
What is the area under the entire normal curve?

Example 24

easy
Adult IQ scores are designed with μ=100\mu = 100, σ=15\sigma = 15. About what percent lies between 8585 and 115115?

Example 25

easy
For normally distributed test scores with μ=80\mu = 80, σ=10\sigma = 10, what value is 11 SD below the mean?

Example 26

medium
For a normal distribution with μ=60\mu = 60, σ=8\sigma = 8, between which two values does about 95%95\% of the data lie?

Example 27

medium
For μ=100\mu = 100, σ=15\sigma = 15, find the z-score of x=130x = 130.

Example 28

medium
For μ=50\mu = 50, σ=4\sigma = 4, what value lies 1.51.5 SDs above the mean?

Example 29

medium
For a normal distribution, about what percent of data lies BELOW the mean minus 11 SD?

Example 30

medium
For μ=70\mu = 70, σ=5\sigma = 5, between which two values does about 99.7%99.7\% of the data lie?

Example 31

medium
In a normal distribution, what fraction of data lies between z=−1z = -1 and z=2z = 2?

Example 32

hard
Two normal curves have the same mean but σA=2,σB=5\sigma_A = 2, \sigma_B = 5. Which is taller at the center?

Example 33

hard
For μ=0,σ=1\mu = 0, \sigma = 1 (standard normal), what value is at the 97.597.5th percentile (approximately)?

Example 34

hard
What is the median of a normal distribution with μ=25\mu = 25, σ=4\sigma = 4?

Example 35

hard
For μ=100\mu = 100, σ=15\sigma = 15, what percent of values are below 7070?

Example 36

hard
A normal distribution has μ=0,σ=2\mu = 0, \sigma = 2. Find P(∣X∣>4)P(|X| > 4).

Example 37

hard
For μ=200\mu = 200, σ=25\sigma = 25, what value lies at the 8484th percentile (approximately)?

Example 38

medium
If a distribution is exactly normal, do its tails ever touch the x-axis?

Example 39

challenge
The Central Limit Theorem states that the sampling distribution of the sample mean approaches what shape as nn grows large?

Example 40

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 41

medium
A normal distribution has mean μ=50\mu = 50 and standard deviation σ=4\sigma = 4. Using the 68-95-99.7 rule, what percentage of values lie between 42 and 58?
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Background Knowledge

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

distribution shapestandard deviation intro