Z-Score & Standard Normal Reference Cheat Sheet

A printable reference covering z-scores, standard normal distribution, percentiles, probability areas, and normal model interpretation for grades 10-12.

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This cheat sheet covers how to use z-scores and the standard normal distribution to compare data values from different normal distributions. Students need these tools to find probabilities, percentiles, and unusual values in statistics problems. A strong understanding of z-scores also helps with sampling distributions and later inference topics. The reference gives a quick way to connect raw scores, standardized scores, and table or calculator probabilities. The main formula is $z = \frac{x - \mu}{\sigma}$ , where $x$ is a data value, $\mu$ is the mean, and $\sigma$ is the standard deviation. A positive z-score is above the mean, a negative z-score is below the mean, and $z = 0$ is exactly at the mean. The standard normal distribution has mean $0$ and standard deviation $1$ . Areas under the normal curve represent probabilities, so $P(a < Z < b)$ is found by subtracting cumulative areas.

Key Facts

The z-score formula is $z = \frac{x - \mu}{\sigma}$ for a population mean $\mu$ and population standard deviation $\sigma$ .
To convert a z-score back to a raw value, use $x = \mu + z\sigma$ .
The standard normal random variable $Z$ has mean $\mu = 0$ and standard deviation $\sigma = 1$ .
A z-score tells how many standard deviations a value is from the mean, so $z = 2$ means $2$ standard deviations above the mean.
For cumulative probability, $P(Z < z)$ is the area to the left of $z$ under the standard normal curve.
To find an interval probability, use $P(a < Z < b) = P(Z < b) - P(Z < a)$ .
By symmetry of the standard normal curve, $P(Z > z) = P(Z < -z)$ .
The empirical rule says about $68\%$ of normal data are within $1\sigma$ , $95\%$ within $2\sigma$ , and $99.7\%$ within $3\sigma$ of the mean.

Vocabulary

Z-score: A z-score is a standardized value that tells how many standard deviations a data value is from the mean.
Standard normal distribution: The standard normal distribution is a normal distribution with mean $0$ and standard deviation $1$ .
Cumulative probability: Cumulative probability is the area under a distribution curve to the left of a given value, written as $P(Z < z)$ .
Percentile: A percentile is the percentage of values in a distribution that are at or below a given value.
Normal curve: A normal curve is a symmetric, bell-shaped curve used to model many real data distributions.
Tail area: A tail area is the probability in the far left or far right end of a distribution beyond a selected cutoff.

Common Mistakes to Avoid

Using $z = \frac{\mu - x}{\sigma}$ instead of $z = \frac{x - \mu}{\sigma}$ is wrong because it reverses the sign and changes whether the value is above or below the mean.
Forgetting to subtract cumulative areas for an interval is wrong because $P(a < Z < b)$ is not the same as $P(Z < b)$ .
Treating a z-score as a probability is wrong because a z-score is a location, while probability is an area under the curve.
Using the standard normal table without checking whether it gives left-tail area is wrong because some tables report different area types.
Rounding z-scores too early is wrong because small rounding changes can noticeably affect probabilities and percentiles.

Practice Questions

1 A test has mean $\mu = 75$ and standard deviation $\sigma = 8$ . Find the z-score for a student who scored $x = 91$ .
2 A normal distribution has $\mu = 120$ and $\sigma = 15$ . What raw value corresponds to $z = -1.6$ ?
3 Using a standard normal table or calculator, find $P(-1.25 < Z < 0.80)$ .
4 A value has $z = 2.4$ in one class and another value has $z = -2.4$ in another class. Explain how their locations compare relative to their own class distributions.

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