Z-Score (Standard Score): Formula, Worked Example, and How to Read the Z-Table

A z-score (also called a standard score) tells you how many standard deviations an observation sits above or below the mean of its distribution. It converts a raw value, which is tied to its own units and scale, into a single dimensionless number that can be compared across variables, instruments, and even entirely different measurement scales.

This guide covers the z-score formula for a population and for a sample, a fully worked example, what standardization does and doesn’t do to a distribution, how to correctly read a standard normal (z) table (including a full table you can use directly), the common critical values built from z, and where z-scores break down.

What a Z-Score Is

A z-score expresses the distance between a single observation and the mean, measured in standard-deviation units:

• A z-score of 0 means the observation equals the mean.
• A positive z-score means the observation is above the mean.
• A negative z-score means the observation is below the mean.
• The magnitude tells you how far away it is: |z| = 2 means the observation is two standard deviations from the mean, regardless of what units the original variable was measured in.

That last point is the whole reason z-scores exist: they are dimensionless. A z-score computed from exam marks and a z-score computed from reaction times in milliseconds are directly comparable, because both have been rescaled onto the same standard-deviation metric. See standard deviation for how σ itself is calculated.

The Z-Score Formula

When the population mean (μ) and population standard deviation (σ) are known:

z = (x − μ) / σ

When you only have a sample, and are estimating the mean and spread from that sample, the sample form uses the sample mean (x̄) and sample standard deviation (s):

z = (x − x̄) / s

Where:

• x = the individual observation being standardized
• μ / x̄ = the mean of the population or sample
• σ / s = the standard deviation of the population or sample

Which version is technically correct matters more than it looks — see z-score vs. t-score below for why using the sample formula and then treating the result as if it came from a known population σ is one of the most common misapplications of this statistic.

Worked Example

Illustrative example, not a real dataset or study. Suppose a set of exam scores has a mean of μ = 75 and a population standard deviation of σ = 8. A student scores x = 87.

z = (87 − 75) / 8 = 12 / 8 = 1.5

That student’s score sits 1.5 standard deviations above the mean. Looking up z = 1.5 in the standard normal table below gives a cumulative area of 0.9332 — meaning roughly 93.3% of scores in this distribution fall at or below 87, so the student scored better than about 93% of the group (assuming the underlying scores are approximately normally distributed — more on that caveat below).

A second illustrative case, this time below the mean: a different observation from a distribution with x̄ = 50, s = 5, and x = 42:

z = (42 − 50) / 5 = −8 / 5 = −1.6

The negative sign means this observation is 1.6 standard deviations below the mean. Its cumulative area from the table is 0.0548 (about the 5.5th percentile).

Standardization and the Standard Normal Distribution

Standardization is the process of converting every value in a dataset to a z-score. The resulting set of z-scores always has a mean of exactly 0 and a standard deviation of exactly 1, no matter what the original data looked like. When the original variable is itself normally distributed, its standardized version is the standard normal distribution — the bell curve with μ = 0 and σ = 1 that the z-table below is built from.

Common misconception: standardizing a non-normal distribution does not make it normal. Z-scores rescale the mean and spread; they do not change the shape of the distribution. A right-skewed distribution is still right-skewed after standardization — only its center and scale have changed. This matters because the percentile and probability statements you can make from a z-table assume the underlying data is approximately normal to begin with. If that assumption doesn’t hold, the z-table’s percentages are misleading. See normality of distribution for how to check that assumption before relying on z-based probabilities, and skewness for what shape distortion looks like.

The Standard Normal (Z) Table

A z-table (also called a standard normal table) gives the cumulative area under the standard normal curve to the left of a given z-score — that is, the proportion of the distribution that falls at or below that z-value. This left-tail, cumulative-from-the-bottom convention is the standard one used by nearly every published z-table and is the convention used below. Misreading which direction a table is oriented is one of the most common z-table errors, so it’s worth stating explicitly before using it.

How to read the table

Find the row matching the z-score to one decimal place, then the column matching the second decimal place, and read the value at their intersection. For example, for z = 1.96: find row “1.9”, column “0.06”, giving 0.9750.

• Area below z (P(Z ≤ z)): read the table value directly. For z = 1.96, that’s 0.9750, or 97.5%.
• Area above z (P(Z > z)): subtract the table value from 1. For z = 1.96: 1 − 0.9750 = 0.0250, or 2.5%.
• Area between two z-values (P(z₁ < Z < z₂)): subtract the smaller table value from the larger one. Between z = 0.5 and z = 1.5: 0.9332 − 0.6915 = 0.2417, or 24.17%.
• Negative z-values: by the symmetry of the normal curve, the area below −z equals 1 minus the area below +z. For z = −1.6: 1 − 0.9452 = 0.0548.

Values are the cumulative area to the left of z (that is, P(Z ≤ z) for the standard normal distribution), rounded to four decimal places. For negative z, use 1 minus the value for the corresponding positive z.

From Z-Scores to Percentiles and Probabilities

Once a distribution is approximately normal, a z-score converts directly into a percentile via the table above: multiply the cumulative area by 100. A z of 1.5 is roughly the 93rd percentile; a z of −1.6 is roughly the 5th percentile.

A useful shortcut for round numbers is the 68-95-99.7 rule (the empirical rule): in an approximately normal distribution, about 68% of values fall within ±1 standard deviation of the mean (z between −1 and 1), about 95% fall within ±2 standard deviations, and about 99.7% fall within ±3 standard deviations. Like every statement in this section, it depends on the data actually being close to normal — see normality of distribution for how to check that before leaning on it.

Common Critical Values

Certain z-scores recur constantly in inferential statistics because they mark standard significance thresholds on the standard normal curve:

• z = 1.645 — the one-tailed 5% critical value. The area to the right of 1.645 is 0.05.
• z = 1.96 — the two-tailed 5% critical value. The area beyond ±1.96 (both tails combined) is 0.05, i.e., 0.025 in each tail — the basis of the standard 95% confidence interval.
• z = 2.576 — the two-tailed 1% critical value, with 0.005 in each tail — the basis of a 99% confidence interval.

These values come directly from the z-table above: they’re simply the z-scores whose cumulative areas correspond to the standard 90%, 95%, and 99% confidence levels. See confidence intervals for how they’re used to build an interval, and what a p-value is for how the same tail areas underpin hypothesis testing.

Z-Score vs. T-Score: Which to Use

Use a z-based approach when the population standard deviation (σ) is genuinely known, or when the sample is large enough (conventionally n ≥ 30) that the sample standard deviation is a reliable stand-in for it. Use a t-based approach when σ is unknown and is being estimated from a small sample — the t-distribution has heavier tails than the standard normal to account for the extra uncertainty that comes from estimating both the mean and the spread from the same limited data. As sample size grows, the t-distribution converges toward the standard normal, which is why the distinction matters most for small samples. See t-test for the full mechanics of when and how each form is applied.

Using Z-Scores to Screen for Outliers

A common rule of thumb flags any observation with |z| > 3 as a potential outlier, since under a normal distribution fewer than 0.3% of values should fall that far from the mean. Treat this as a starting heuristic, not a rule: it depends on the mean and standard deviation themselves being reasonably stable estimates, and both are sensitive to the very outliers you’re trying to detect, which can mask them (a single extreme value inflates the standard deviation and can pull its own z-score back under the threshold). The |z| > 3 rule is also unreliable for small samples and for data that isn’t approximately normal to begin with. For skewed data or smaller datasets, an IQR-based rule is generally more robust — see interquartile range and the five-number summary for that approach.

Where Z-Scores Are Used

• Standardized test scoring — converting raw scores onto a common scale so performance is comparable across test forms or administrations.
• Clinical and growth reference scores — several clinical measurements are reported as standardized scores against a reference population (for example, bone density results are commonly reported as T-scores and Z-scores relative to different reference groups). The specific clinical cut-offs used for any such measure are set by the relevant clinical guidelines, not by the statistical method itself, and are outside the scope of this guide.
• Statistical process control / quality control — flagging measurements that fall an unusual number of standard deviations from a process mean.
• Standardizing predictors before regression — converting predictor variables to z-scores (mean 0, SD 1) so their coefficients become comparable in magnitude, and so interaction and polynomial terms are less prone to multicollinearity. See regression analysis.

Common Errors

• Using the sample standard deviation as if it were the population standard deviation. The two formulas above look almost identical, but conflating them overstates your confidence in the result, especially with small samples — this is exactly the distinction the z vs. t section above addresses.
• Misreading the table’s direction. Always confirm whether a given table reports cumulative area to the left (the convention used here) or something else before reading off a value — tables from different textbooks are not always laid out the same way.
• Applying z-based probabilities to skewed or non-normal data. The percentages in the table above are only meaningful once you’ve confirmed the underlying distribution is approximately normal.
• Confusing a z-score with a z-test. A z-score standardizes a single observation (or a single sample mean). A z-test uses that standardized value, compared against a critical value, to make a formal hypothesis-testing decision. The score is an input to the test, not the test itself.

Frequently Asked Questions

What is the z-score formula?

For a population: z = (x − μ) / σ. For a sample: z = (x − x̄) / s. Both divide the distance between an observation and the mean by the standard deviation, producing a dimensionless standardized value.

How do you read a z-score table?

Find the row for the z-value to one decimal place and the column for the second decimal place; their intersection is the cumulative area to the left of that z-score (the proportion of the distribution at or below it). Subtract from 1 for the area above, or subtract two table values for the area between two z-scores.

What does a negative z-score mean?

The observation is below the mean. Its distance from the mean, in standard-deviation units, is given by the absolute value of z; the sign only indicates direction.

What’s the difference between a z-score and a z-test?

A z-score standardizes one value. A z-test is a hypothesis test that compares a standardized statistic (often a sample mean’s z-score) against a critical value to decide whether to reject a null hypothesis.

When should I use a z-score instead of a t-score?

Use z when the population standard deviation is known or the sample is large (n ≥ 30 is the usual rule of thumb). Use t when the standard deviation is estimated from a small sample, since the t-distribution’s heavier tails account for that added uncertainty.

Does standardizing a dataset make it normally distributed?

No. Standardization rescales the mean to 0 and the standard deviation to 1, but it does not change the shape of the distribution. A skewed dataset is still skewed after standardization.