What does a negative z-score mean?

A negative z-score means the data point is below the mean. For example, if the mean score is 75 and a student scores 65 with a z-score of -1, the student is 1 standard deviation below the mean. The negative sign just indicates direction; it doesn't mean anything is wrong.

What is the difference between z-score and z-table?

A z-score is the standardized value you calculate from your data using the formula z = \frac{x - \mu}{\sigma}. A z-table lists pre-calculated probabilities and percentiles for each z-score in a standard normal distribution. After computing the z-score, you look it up in a z-table to find what percent of data falls below that point.

Can a z-score be greater than 3?

Yes, a z-score can be any real number. However, in a normally distributed dataset, z-scores beyond ±3 are extremely rare (less than 1 in 1000 chance) and almost always indicate an outlier or unusual value.

How do I use a z-score to find probabilities?

After calculating the z-score, use a z-table, calculator, or software to look up the cumulative probability (percentile) for that z-score. For example, z = 1.96 corresponds to the 97.5th percentile, meaning 97.5% of data in a normal distribution falls below that point.

Z-Score

The Z-Score tells you how many standard deviations a data point is from the mean, revealing whether it is typical or unusual.

z = \frac{x - \mu}{\sigma}

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What each symbol means

What Z-Score takes

Symbol	Meaning
$z$	The standardized score (unitless), showing how many standard deviations $x$ is from the mean; if misread as a raw value instead of a comparison tool, you lose the ability to identify outliers.
$x$	The individual data point being measured, in the original units; if confused with $z$ , you lose standardization and cannot compare across different distributions.
$\mu$	The population mean (average) in the original units; if confused with the median, you may choose the wrong measure of center for a skewed distribution.
$\sigma$	The population standard deviation (always positive), measuring spread; if mistaken for sample standard deviation, your z-score will be incorrect.

When to use it

Use z-score when you need to standardize a data point and see how it compares to the rest of the dataset.

Level

Usually taught in: Statistics · Appears on: AP Statistics

Worked examples

1. Calculate the z-score for a test score above the mean

Problem

A student scored 85 on a test where the class mean is 75 and the standard deviation is 5. What is the z-score?

$z = \frac{x - \mu}{\sigma}$
Recall the z-score formula: subtract the mean from the data point, then divide by the standard deviation.
$z = \frac{85 - 75}{5}$
Substitute the values: $x = 85$ (the test score), $\mu = 75$ (the class mean), and $\sigma = 5$ (the standard deviation).
$z = \frac{10}{5}$
Compute the numerator: $85 - 75 = 10$ .
$z = 2$
Divide to get the final z-score: $10 \div 5 = 2$ .

Answer: $z = 2$

This z-score tells us the student scored 2 standard deviations above the class mean. In a normal distribution, a z-score of 2 is a strong performance—better than about 97.7% of the class.

2. Calculate the z-score for a value below the mean with decimal inputs

Problem

A bottle factory measures the diameter of bottle caps, which should average 20 mm with a standard deviation of 0.8 mm. One cap measures 18.4 mm. What is its z-score?

$z = \frac{x - \mu}{\sigma}$
Write out the z-score formula.
$z = \frac{18.4 - 20}{0.8}$
Substitute: $x = 18.4$ mm (the cap diameter), $\mu = 20$ mm (the target mean), and $\sigma = 0.8$ mm (the standard deviation).
$z = \frac{-1.6}{0.8}$
Compute the numerator: $18.4 - 20 = -1.6$ (the cap is undersized).
$z = -2$
Divide: $-1.6 \div 0.8 = -2$ (negative because the cap is below the mean).

Answer: $z = -2$

A z-score of -2 means this cap is 2 standard deviations below the target diameter. In quality control, caps this far from the mean would typically be flagged for inspection or rejection.

3. Find the z-score for a height in a population

Problem

The average height of adult women in a country is 64 inches with a standard deviation of 2.5 inches. A woman is 69 inches tall. What is her z-score?

$z = \frac{x - \mu}{\sigma}$
Recall the z-score formula; we need to standardize the woman's height against the population mean and standard deviation.
$z = \frac{69 - 64}{2.5}$
Identify the values: $x = 69$ inches (the woman's height), $\mu = 64$ inches (the average height), and $\sigma = 2.5$ inches (the standard deviation).
$z = \frac{5}{2.5}$
Subtract the mean from the height: $69 - 64 = 5$ inches.
$z = 2$
Divide by the standard deviation: $5 \div 2.5 = 2$ .

Answer: $z = 2$

A z-score of 2 means this woman is 2 standard deviations taller than the average woman. This is unusually tall; she is taller than roughly 97.7% of women in the population. Without the z-score, it is hard to know whether 69 inches is truly exceptional without understanding the typical spread of heights.

Common mistakes

Where Z-Score usually goes wrong

The mistake	Why it is wrong	The fix
Writing $z = \frac{x - \sigma}{\mu}$ (reversing the mean and standard deviation).	This formula reverses the roles of the parameters and doesn't measure deviations from the mean, producing a meaningless result.	Always use $z = \frac{x - \mu}{\sigma}$ : subtract the mean first, then divide by the standard deviation.
Using sample standard deviation ( $s$ ) when the problem specifies population parameters.	The sample standard deviation has a slightly different value than the population standard deviation, so the z-score will be inaccurate, especially for small samples.	Check whether the data is the whole population (use $\sigma$ ) or a sample (use $s$ ); clarify which standard deviation applies before calculating.
Reporting the z-score with the original units (e.g., 'inches' or 'dollars').	The z-score is a standardized, unitless measure; attaching units incorrectly suggests it is a raw measurement and destroys its meaning as a comparison tool.	Always report the z-score as a plain number with no units, and explain what it means in terms of standard deviations from the mean.

Tips and when to use something else

z-scores are most useful for approximately normally distributed data; for heavily skewed distributions, try the Empirical Rule or Outlier Rule instead.
A z-score between -2 and 2 typically contains about 95% of a normal distribution; any z-score beyond ±3 is almost certainly an outlier.
To compare data from two different datasets, calculate z-scores for both using their own means and standard deviations so you compare apples to apples.
z-scores tell you distance from the mean, not probability; use a z-table or Normal Distribution calculator to convert a z-score to a percentile or probability.

Related formulas

Standard DeviationStandard deviation measures how spread out data points are from the average—use it to understand variability in a dataset.Normal DistributionThe normal distribution describes the probability density of a continuous variable, used to model measurements and natural phenomena.Empirical RuleThe Empirical Rule describes where approximately 68%, 95%, and 99.7% of normally distributed data fall, measured in standard deviations from the mean.Outlier RuleThe Outlier Rule identifies data values farther than 1.5 times the interquartile range from the quartiles—use it to flag unusual observations.

Frequently asked questions

What does a negative z-score mean?: A negative z-score means the data point is below the mean. For example, if the mean score is 75 and a student scores 65 with a z-score of -1, the student is 1 standard deviation below the mean. The negative sign just indicates direction; it doesn't mean anything is wrong.
What is the difference between z-score and z-table?: A z-score is the standardized value you calculate from your data using the formula $z = \frac{x - \mu}{\sigma}$ . A z-table lists pre-calculated probabilities and percentiles for each z-score in a standard normal distribution. After computing the z-score, you look it up in a z-table to find what percent of data falls below that point.
Can a z-score be greater than 3?: Yes, a z-score can be any real number. However, in a normally distributed dataset, z-scores beyond ±3 are extremely rare (less than 1 in 1000 chance) and almost always indicate an outlier or unusual value.
How do I use a z-score to find probabilities?: After calculating the z-score, use a z-table, calculator, or software to look up the cumulative probability (percentile) for that z-score. For example, $z = 1.96$ corresponds to the 97.5th percentile, meaning 97.5% of data in a normal distribution falls below that point.

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Reviewed 2026-09-18