A Z-Score Calculator computes z-score from the inputs you provide. It applies the standard formula to the values you enter and returns the result instantly, without sending any data to a server. Free Z-Score Calculator. The tool runs entirely in.
Standardize a value against a normal distribution. z = (x - mu) / sigma.
Formula: z = (x - mu) / sigma. The cumulative normal probability uses the Abramowitz and Stegun 26.2.17 polynomial approximation. The two-tailed p-value is 2 x (1 - Phi(|z|)). Source: NIST Handbook of Mathematical Functions.
Z = (x - μ) / σ - universal scoring across normal distributions.
A z-score expresses any value as 'how many standard deviations from the mean.' It standardizes scores across distributions (test scores, body measurements, stock returns). 95% of normally-distributed data falls within ±2 standard deviations.
Enter your value, the population mean, and the standard deviation. The calculator subtracts the mean from the value, divides by the standard deviation, and gives the percentile assuming a normal distribution.
| Measure | Formula | Best when |
|---|---|---|
| Mean (arithmetic average) | sum / n | Symmetric data, no outliers |
| Median | Middle value when sorted | Skewed data (income, house prices) |
| Mode | Most frequent value | Categorical data |
| Range | max - min | Quick spread; sensitive to outliers |
| Variance | Σ(xi - mean)² / n | Spread; in squared units |
| Standard deviation (σ) | √Variance | Spread in original units |
| IQR (interquartile range) | Q3 - Q1 | Robust spread, ignores outliers |
The bell curve. Most natural measurements (heights, IQ, exam scores) approximate normal. Key properties:
A z-score expresses how many σ a value is from the mean: z = (x - mean) / σ. z = 1.96 corresponds to the 97.5 percentile (95% confidence interval bound).
| Test | Use when | Tells you |
|---|---|---|
| t-test (one sample) | Compare one mean to a known value | Is sample mean different from this number? |
| t-test (independent) | Compare means of two groups | Are these two groups different? |
| t-test (paired) | Compare same subjects before/after | Did treatment change the outcome? |
| ANOVA | Compare means of 3+ groups | Is at least one group different? |
| Chi-square | Categorical data (e.g., 2x2 tables) | Is there association between categories? |
| Pearson correlation | Linear relationship between 2 continuous variables | Strength and direction (-1 to +1) |
| Linear regression | Predict one variable from another | Slope, intercept, R^2 fit |
The p-value is the probability of seeing data this extreme (or more) IF the null hypothesis were true. Common misuses:
For estimating a proportion within ±3 percentage points (e.g., a poll):
This is why political polls cluster around n=1,000.
This calculator uses the following formula:
z = (x - μ) / σ - positive means above the mean, negative below
The reason this formula works is rooted in the underlying physics, finance, or biology of the problem. Behind every calculator is a published, peer-reviewed equation or a widely accepted convention. We do not invent formulas; we apply standard ones from textbooks, government tables, professional bodies, and academic literature.
If you are curious about the math, the simplest way to verify is to plug in two known numbers and compare against a known result. The calculator should match published examples to within rounding precision.
Depends on context. For test scores, +1 to +2 is 'above average to outstanding'. For quality control, |z| > 3 flags an outlier.
Z assumes you know population standard deviation. T-score uses sample standard deviation and is used for small samples (n < 30).
Only relative to the mean. Z = -2 on cholesterol is good (low). Z = -2 on GPA is bad. Direction matters.
z = 0 is the 50th percentile (median). z = +1 is ~84th, z = -1 is ~16th. Use the empirical rule: 68% within ±1, 95% within ±2.
No. Correlation = two things move together. Causation = one causes the other. Confounders, reverse causation, and coincidence can all create correlation without causation. To establish causation: randomized controlled trials, natural experiments, or instrumental variables.
Field-dependent. In physics, r = 0.95 is normal; in social science, r = 0.4 is strong. Squared r (R^2) tells you the % of variance explained: r = 0.7 means ~49% of variance is shared.
If data is symmetric (no skew, no outliers): mean. If skewed (income, house prices, response times): median. Always report both for unfamiliar audiences plus a measure of spread (σ for symmetric, IQR for skewed).
Depends on effect size, variability, and desired power. For detecting a medium effect (Cohen's d = 0.5) with 80% power at α = 0.05: ~64 per group for a t-test. Use a power calculator before running the study.
t-test assumes normal data and equal variances. Mann-Whitney (Wilcoxon) is non-parametric - it ranks values and doesn't assume distribution. For non-normal data, Mann-Whitney is more robust but slightly less powerful.
It applies the standard formula. Accuracy is limited only by your input precision. For decisions with material consequences (taxes, medical, legal, structural), use the result as a starting point and verify with a qualified professional in the relevant field.
Yes. 100% free, no signup, no payment, no API key. The site is funded by display ads around the tool but not inside the calculation flow.
No. All inputs stay in your browser tab. Closing the tab discards them. The site uses Google Analytics for traffic measurement (anonymized) but the analytics never see what you type into the form.
Yes. The tool is responsive and tested on iOS Safari, Android Chrome, and major desktop browsers. Touch targets meet Apple's 44pt and Google's 48dp minimum.
Yes. Once the page has loaded, it works without internet. The calculation runs in JavaScript on your device.
Email hi@3tej.com with the URL of this page and a description of what you saw vs expected. We typically respond within 72 hours.
Take a screenshot or copy the output. The page doesn't generate shareable URLs for specific calculations - inputs stay in your browser only.
Most likely: different formula assumptions, different default values, different rounding rules, or different applicable rates. Check the methodology if both tools document it. Both can be valid for different scenarios.