Normal Distribution & Empirical Rule Cheat Sheet

A printable reference covering normal distributions, z-scores, standard deviation, the Empirical Rule, percentiles, and probabilities for grades 9-12.

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A normal distribution is a bell-shaped model used to describe many real data sets, such as test scores, heights, and measurement errors. This cheat sheet helps students connect the shape of the curve to mean, standard deviation, z-scores, and probability. It is especially useful for quickly estimating how unusual a value is and what percent of data falls in a given interval. The most important ideas are that the mean $\mu$ is at the center, the standard deviation $\sigma$ controls spread, and total area under the curve equals $1$ . The Empirical Rule says about $68\%$ of values fall within $1$ standard deviation, $95\%$ within $2$ , and $99.7\%$ within $3$ . A z-score, $z = \frac{x - \mu}{\sigma}$ , converts any normal value into its number of standard deviations from the mean. Percentiles and probabilities come from areas under the normal curve.

Key Facts

A normal distribution is symmetric, bell-shaped, and centered at the mean $\mu$ .
In a normal distribution, the mean, median, and mode are all equal: $\text{mean} = \text{median} = \text{mode}$ .
The total area under a normal curve is $1$ , which represents $100\%$ of the data.
The Empirical Rule says about $68\%$ of data lies between $\mu - \sigma$ and $\mu + \sigma$ .
The Empirical Rule says about $95\%$ of data lies between $\mu - 2\sigma$ and $\mu + 2\sigma$ .
The Empirical Rule says about $99.7\%$ of data lies between $\mu - 3\sigma$ and $\mu + 3\sigma$ .
A z-score is calculated with $z = \frac{x - \mu}{\sigma}$ and tells how many standard deviations $x$ is from the mean.
For any normal distribution, standardizing with $z = \frac{x - \mu}{\sigma}$ changes it to the standard normal distribution with $\mu = 0$ and $\sigma = 1$ .

Vocabulary

Normal distribution: A symmetric bell-shaped distribution where most values are near the mean and fewer values occur farther away.
Mean: The center or balance point of a normal distribution, usually written as $\mu$ .
Standard deviation: A measure of spread, written as $\sigma$ , that describes how far values typically are from the mean.
Empirical Rule: A rule for normal distributions stating that about $68\%$ , $95\%$ , and $99.7\%$ of data fall within $1$ , $2$ , and $3$ standard deviations of the mean.
Z-score: A standardized value that tells how many standard deviations a data value is above or below the mean.
Percentile: A location in a distribution showing the percent of data values at or below a given value.

Common Mistakes to Avoid

Using the Empirical Rule for non-normal data is wrong because the $68\%$ , $95\%$ , and $99.7\%$ pattern only applies well to bell-shaped, approximately normal distributions.
Forgetting that standard deviation must be positive is wrong because $\sigma$ measures spread and cannot be less than $0$ .
Subtracting in the wrong order for a z-score is wrong because the correct formula is $z = \frac{x - \mu}{\sigma}$ , not $z = \frac{\mu - x}{\sigma}$ .
Thinking a negative z-score means an impossible value is wrong because a negative z-score only means the value is below the mean.
Confusing area with height on the curve is wrong because probability is represented by area under the curve, not by how tall the curve is at one point.

Practice Questions

1 A normal distribution has mean $\mu = 80$ and standard deviation $\sigma = 5$ . Find the interval that contains about $95\%$ of the data.
2 A test score of $x = 72$ comes from a normal distribution with $\mu = 60$ and $\sigma = 8$ . Calculate the z-score using $z = \frac{x - \mu}{\sigma}$ .
3 In a normal distribution with $\mu = 100$ and $\sigma = 15$ , estimate the percent of data between $70$ and $130$ .
4 Explain why two data values with the same z-score from different normal distributions have the same relative position, even if the original values are different.

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