Z-Score Calculator
Convert any raw value into a z-score using the mean and standard deviation, or let the calculator derive them from your own data set. You also get the percentile and tail probabilities.
Calculate Your Z-Score
Z-Score Formula
Where:
- \(x\) — the raw value you want to standardize
- \(\mu\) — the mean of the population (or of your data set)
- \(\sigma\) — the standard deviation of the population (or the sample standard deviation, \(s\))
- \(z\) — the z-score, i.e. the number of standard deviations between \(x\) and the mean
The formula measures the distance from the value to the mean and expresses it in units of standard deviation. A positive z means the value is above the mean, a negative z means it is below.
Z-Score – Calculation Example
Example: an exam is scored with a mean of \(\mu = 100\) and a standard deviation of \(\sigma = 15\). A student scores \(x = 110\). How many standard deviations above the mean is that, and what percentile does it correspond to?
\[ z = \frac{x - \mu}{\sigma} = \frac{110 - 100}{15} = \frac{10}{15} \approx 0.6667 \]
\[ \Phi(0.6667) \approx 0.7475 \quad\Rightarrow\quad P(X \le 110) \approx 74.75\% \]
Result: \(z \approx 0.6667\). The score is about two-thirds of a standard deviation above the mean; roughly 74.75% of students score at or below 110, and about 25.25% score higher.
How to use the calculator
- Open the Input mode dropdown and choose Mean & standard deviation known if you already have \(\mu\) and \(\sigma\), or From a data set if you want the calculator to compute them for you.
- In Raw value (x), enter the value you want to convert into a z-score.
- In direct mode, fill in Population mean (μ) and Standard deviation (σ). The standard deviation must be greater than 0.
- In data-set mode, paste your numbers into Data set, separated by commas, spaces, semicolons or line breaks. You need at least 2 numbers, and they cannot all be identical. The calculator computes the mean and the sample standard deviation (using \(n - 1\)) automatically.
- Read the Z-score, the percentile \(P(X \le x)\), the probability of a higher value \(P(X > x)\), and a plain-language Interpretation with a summary table.
What a z-score means
A z-score (also called a standard score) answers a simple question: how unusual is this value? It converts any measurement into a common scale by asking how many standard deviations it sits away from the mean. Because every distribution's raw units are stripped away, z-scores let you compare values from completely different measurements — for example, whether a height of 190 cm or an exam score of 130 is more exceptional relative to its own population.
The sign carries the direction. A z-score of +2 means the value is two standard deviations above the mean; a z-score of −1.5 means it is one and a half standard deviations below. A z of exactly 0 means the value equals the mean.
The calculator also converts your z-score into probabilities, assuming the data follow a normal (Gaussian) distribution. The percentile \(\Phi(z)\) is the share of values expected at or below yours; \(1 - \Phi(z)\) is the share expected above. For the example above, \(z = 0.6667\) places the value at about the 75th percentile — a good but not exceptional result.
Interpreting how extreme a z-score is
The familiar 68–95–99.7 rule for normal distributions gives quick benchmarks:
| |z| range | What it means | Share of values at least this extreme |
|---|---|---|
| under 1 | Typical — within 1 standard deviation of the mean | ≈ 32% |
| 1 to 2 | Somewhat uncommon | ≈ 27% |
| 2 to 3 | Unusual | ≈ 4.6% |
| 3 or more | Very unusual | under 0.3% |
Z-scores beyond ±3 are often treated as outliers or as signals worth investigating, though what counts as "extreme" depends on your field — in physics, 5-sigma (\(|z| \ge 5\)) is a common threshold for a discovery.
Assumptions, limitations and tips
- Percentiles assume normality. The z-score itself is valid for any distribution with a defined mean and standard deviation, but the percentile and tail probabilities \(\Phi(z)\) are only accurate when the data are approximately normally distributed. For skewed data, the reported probabilities will be misleading even though the z-score is correct.
- Sample vs. population standard deviation. In data-set mode the calculator uses the sample standard deviation with Bessel's correction (dividing by \(n - 1\)), which is the right choice when your numbers are a sample from a larger population. If your numbers are the entire population, enter \(\sigma\) manually in direct mode instead.
- Small samples are noisy. With only a handful of values, the sample mean and standard deviation are unstable estimates, so z-scores computed from them should be treated with caution.
- Watch for outliers at the input stage. A single mistyped value in the data set can inflate the standard deviation and flatten everyone else's z-scores. Double-check the summary table in the interpretation block — it shows the mean, standard deviation and sample size that were actually used.
- Use the raw value, not a z-score, in the x field. The calculator expects an original measurement; it computes the standardization for you.
Frequently Asked Questions
How do I calculate a z-score by hand without a calculator?
Subtract the mean from your value, then divide by the standard deviation: \(z = (x - \mu) / \sigma\). For example, with \(x = 85\), \(\mu = 70\) and \(\sigma = 10\), you get \(z = 15/10 = 1.5\). If you don't know \(\mu\) and \(\sigma\), first compute the mean, then the standard deviation, and plug them into the formula. Doing it by hand is a good sanity check on the calculator's result.
What is a good or normal z-score?
There is no universal "good" — it depends on what you're measuring. Statistically, most values in a normal distribution fall between −1 and +1 (about 68%), so anything in that range is typical. Scores beyond ±2 or ±3 are rare. Whether high is good depends on context: a high z-score is great for an exam but may be a warning sign for blood pressure. For children's growth charts, doctors look at where the z-score falls between roughly −2 and +2; always ask a professional about medical results.
Does a z-score have units? Can I compare z-scores from different measurements?
No — a z-score is dimensionless. It expresses distance from the mean in units of standard deviations, so the original units (cm, kg, points, dollars) cancel out. That's exactly what makes z-scores useful for comparison: a z-score of 1.8 for a height and a z-score of 1.2 for an exam score tell you the height is more exceptional relative to its own population, even though the raw numbers have nothing in common.
How do I convert a z-score back to a raw value?
Rearrange the formula: \(x = \mu + z \times \sigma\). For instance, if \(z = -1.5\), \(\mu = 200\) and \(\sigma = 20\), then \(x = 200 + (-1.5)(20) = 170\). This is handy when you know a cut-off in z terms — for example, "values below z = −2 are flagged" — and need the corresponding raw threshold in the original units.
Why does my z-score differ when I compute it from a data set instead of entering σ directly?
In data-set mode the calculator uses the sample standard deviation with Bessel's correction — dividing by \(n - 1\) instead of \(n\) — which slightly inflates the spread estimate for small samples. If you compared that against a population \(\sigma\) (dividing by \(n\)), you'll get a different z for the same raw value. Both are defensible; the difference shrinks as your sample grows, but it can be noticeable with fewer than about 10 numbers.
Can a z-score be negative, and what does a negative z-score mean?
Yes. A negative z-score simply means the value lies below the mean — nothing more. A z of −1.25 says the value is 1.25 standard deviations under the average. Negative z-scores are perfectly normal in symmetric data: about half of any normally distributed population has them. Only the absolute value tells you how unusual a result is, so −2.1 is just as extreme as +2.1.
What's the difference between a z-score and a t-score or a p-value?
A z-score is the standardized distance from the mean in standard deviation units. A t-score is a rescaled version common in standardized testing (e.g., bone density): \(t = 50 + 10z\), so the average is 50 instead of 0. A p-value is a probability, not a distance — often the two-tailed probability associated with a z or t statistic. They're related but answer different questions, so don't interchange them.