Z-Score Calculator
Convert a raw value to a z-score and probability, find the value at a target percentile, or find the probability between two z-scores.
How to use this calculator
- 1Choose what you want to solve for: a z-score and probability from a raw value, a value and z-score from a target percentile, or the probability between two z-scores.
- 2Enter the mean and standard deviation of the distribution, plus whichever other value the mode needs.
- 3Use the two-tailed probability when you care about being unusual in either direction, not just one.
How the calculation works
z = (x − μ) / σ x = μ + zσ P(Z1 < Z < Z2) = Φ(Z2) − Φ(Z1)- x
- The raw value you are testing
- μ
- The mean of the distribution
- σ
- The standard deviation
- Φ
- The standard normal cumulative distribution function — the percentile a given z-score sits at
A z-score expresses distance from the mean in units of standard deviation, which makes values from different distributions directly comparable.
Probabilities are computed from the standard normal cumulative distribution function, using an approximation of the error function accurate to about 1.5 × 10⁻⁷.
Solving for a value from a target percentile runs the same relationship backwards, via the inverse of that same function (Acklam's rational approximation).
Worked example
A score of 85 where the mean is 70 and σ is 10
- 1.z = (85 − 70) ÷ 10 = 1.5.
- 2.The value sits one and a half standard deviations above the mean.
- 3.The standard normal CDF at 1.5 is 0.9332, so this is the 93.32nd percentile.
- 4.Only about 6.68% of the distribution scores higher.
Result: z = +1.5, the 93.3rd percentile
The probability between z = −1 and z = +1
- 1.Φ(1) = 0.8413 and Φ(−1) = 0.1587.
- 2.P(−1 < Z < 1) = 0.8413 − 0.1587 = 0.6827.
- 3.This is the well-known "about 68% of values fall within one standard deviation" figure.
Result: 68.27%
What a z-score actually is
A z-score restates a raw value as a distance from the mean, measured in standard deviations rather than in the original units. A z-score of +1.5 means a value sits one and a half standard deviations above the mean; −2 means two standard deviations below it. Stripping away the original scale is exactly what makes the transformation useful — a test score, a height and a temperature reading all become directly comparable once each is expressed the same way.
The formula itself is just subtraction followed by division: take the value, subtract the mean, then divide by the standard deviation. Running it backwards — multiplying a z-score by the standard deviation and adding the mean back — recovers the original value, which is how this calculator can also work from a target percentile back to a value.
From z-score to probability
Once a value is expressed as a z-score, the standard normal distribution can translate it into a probability or a percentile — but only if the underlying data is roughly bell-shaped. Under that assumption, a z-score of 0 sits at the 50th percentile, +1 near the 84th, and +2 near the 98th, following the familiar curve that is symmetric and tapers off in both tails.
The widely quoted 68–95–99.7 rule falls straight out of this: about 68% of a normal distribution lies within one standard deviation of the mean, about 95% within two, and about 99.7% within three. The reference table above applies exactly that curve to your own mean and standard deviation.
Where z-scores get used
Because it standardizes anything measured on a roughly normal scale, the z-score turns up across a wide range of fields.
- Standardized testing — converts raw scores from different test forms or years onto one comparable scale.
- Growth charts and medical screening — flags a measurement — height, weight, a lab value — as unusually high or low for a reference group.
- Quality control — treats a process running several standard deviations from its target average as out of control rather than normal variation.
- Finance — measures how unusual a price move or return is relative to its own history.
- Outlier detection — a common rule of thumb flags any point beyond about three standard deviations for closer inspection.
What a z-score does not tell you
A z-score always exists, regardless of the shape of the underlying distribution — it is just arithmetic. The probability and percentile attached to it are a different matter: they are only accurate when the data is genuinely close to normal. Apply the standard normal curve to strongly skewed data — incomes, wait times, most real-world "count" data — and the resulting percentile can be seriously wrong even though the z-score itself was computed correctly.
A z-score also says nothing about whether a value is good or bad — only how unusual it is. A high z-score is a great sign on a test score and a warning sign on a defect rate; the number is neutral, and the meaning comes entirely from what is being measured.
What this assumes, and where it stops
Assumptions
- Percentiles and probabilities assume a normal distribution. The z-score itself does not.
- The mean and standard deviation are known population parameters. If they are sample estimates from a small sample, a t-distribution is more appropriate.
Limitations
- For skewed or heavy-tailed data, normal-based percentiles can be badly wrong, particularly in the tails.
- With samples under about 30, use a t-score rather than a z-score.
Common questions
What is a good z-score?
It depends entirely on context. In a test score, higher is better and +2 is excellent. In a quality-control measurement, any large absolute value signals a problem. The z-score measures unusualness, not quality — the sign and interpretation come from what you are measuring.
When should I use a t-score instead?
When you are estimating the standard deviation from a small sample, typically fewer than 30 observations. The t-distribution has heavier tails to account for the extra uncertainty in that estimate, giving more conservative probabilities.
How do I find the value at a given percentile, like the 90th?
Switch "Solve for" to "Z-score and value, from a target percentile" and enter 90. This runs the usual z-score calculation backwards: instead of converting a known value into a percentile, it converts a target percentile into a z-score and then a value, using the mean and standard deviation you provide.
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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