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Z-Score Calculator

Standardises a value against a mean and standard deviation (z = (x − μ) / σ) and reports the corresponding percentile and tail probabilities under the normal distribution.

When to use

You need how many standard deviations a value lies from the mean, or the percentile a value corresponds to in a normally distributed population (test scores, measurements, quality control).

Do not use when: You need the probability between two values or the density (use normal-distribution), the data are clearly not normal, or you need the empirical percentile rank within an actual data set (use percentile).

Formula

z = (value − mean) / std_dev; percentile = 100 × Φ(z), where Φ is the standard normal CDF

Φ is evaluated with the Abramowitz & Stegun 7.1.26 erf approximation (absolute error below 1.5·10⁻⁷). The percentile interpretation assumes the reference population is normally distributed.

Inputs

Parameter	Type	Required	Description
`value`	number	yes	The observation to standardise.
`mean`	number	yes	Mean of the population or reference distribution.
`std_dev`	number	yes	Standard deviation of the population or reference distribution. Range: > 0

Outputs

Output	Type	Unit	Description
`z_score`	number		(value − mean) / std_dev: distance from the mean in standard deviations.
`percentile`	number	%	Percentage of a normal population with values below x: 100 × Φ(z).
`probability_below`	number		Standard normal CDF Φ(z).
`probability_above`	number		1 − Φ(z).
`interpretation`	string		Plain-language position relative to the mean.

Example

x = 1.96 on the standard normal: {"value":1.96,"mean":0,"std_dev":1} → {"z_score":1.96,"percentile":97.5,"probability_below":0.975,"probability_above":0.025}

Score 85, mean 70, SD 10: {"value":85,"mean":70,"std_dev":10} → {"z_score":1.5,"percentile":93.32,"probability_below":0.9332,"probability_above":0.0668,"interpretation":"1.5 standard deviations above the mean"}

GET https://tttkmbb.com/api/v1/calculate/z-score?value=1.96&mean=0&std_dev=1

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/z-score (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/z-score · Markdown: https://tttkmbb.com/statistics/z-score.md · JSON definition: https://tttkmbb.com/statistics/z-score.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="z-score"
OpenAPI operationId: calculate_z_score
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Standard score (reference)
NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.1 Normal Distribution (government)

FAQ

Should I use the sample or population standard deviation?

Use the population σ when it is known (e.g. a standardised test). When only a sample is available, the sample s is the usual substitute and the percentile becomes approximate, especially for small samples.

What is a 'significant' z-score?

By convention |z| > 1.96 corresponds to the outer 5 % of a normal distribution (two-sided) and |z| > 2.576 to the outer 1 %; values beyond |z| = 3 are often flagged as outliers.

Related calculators

Normal Distribution Calculator — Probabilities between two values and the density at a point.
Percentile Calculator — Empirical percentile rank within an actual list of values.
Descriptive Statistics Calculator — Compute the mean and standard deviation from data first.