The Normal Distribution and Z-Scores Made Simple

The normal distribution is the symmetric bell-shaped curve that describes many real measurements. A z-score says how many standard deviations a value is from the mean: $z = \frac{x - \mu}{\sigma}$. Converting to z-scores lets you find any normal probability from one standard curve.

The bell curve

A normal distribution is fixed by two numbers: the mean $\mu$ (the centre) and the standard deviation $\sigma$ (the spread). It is symmetric, so half the values lie below the mean.

The 68–95–99.7 rule

Example: IQ scores have $\mu = 100$ and $\sigma = 15$, so about 68% of people score between 85 and 115.

Z-scores

Example: a test has mean 70 and standard deviation 8. A score of 82 has

so it is one and a half standard deviations above average. About 93.3% of scores are below it.

Finding probabilities

Probabilities are areas under the curve. The standard normal CDF, written $\Phi(z)$, gives the area to the left of $z$.

• $P(X < a) = \Phi(z_a)$
• $P(X > a) = 1 - \Phi(z_a)$
• $P(a < X < b) = \Phi(z_b) - \Phi(z_a)$

Worked example: adult heights have $\mu = 175$ cm and $\sigma = 7$ cm. What share are taller than 190 cm?

So about 1.6% of adults are over 190 cm.

A famous value: $\Phi(1.96) \approx 0.975$, so 95% of a normal distribution lies between $z = -1.96$ and $z = 1.96$. This is where 95% confidence intervals come from.

When is data normal?

Many measurements are roughly normal: heights, test scores, measurement errors. Others are not: incomes and waiting times are usually skewed. Check a histogram before assuming normality.

Common mistakes

• Forgetting to subtract from 1 for "greater than" questions.
• Using the variance instead of the standard deviation in the z formula.
• Assuming any data set is normal.

Practice questions

With $\mu = 50$ and $\sigma = 10$:

1. Find the z-score of 65.
2. Roughly what percentage lies between 40 and 60?
3. Is a value of 85 unusual?

Answers: 1) $1.5$ 2) about 68% 3) Yes, $z = 3.5$

The normal distribution calculator shades the area under the curve for any range, and the z-score calculator gives the percentile. To find $\sigma$ from data, see standard deviation explained.

Frequently asked questions

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