A z-score (also called standard score) measures how many standard deviations a data point is from the mean of its distribution. Formula: z = (x − μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. Positive z-scores are above the mean; negative z-scores are below.

What is a 'normal' z-score?

For normally distributed data, 68.27% of values have z-scores between −1 and +1, 95.45% between −2 and +2, and 99.73% between −3 and +3. Values with |z| > 2 are often considered unusual; |z| > 3 is considered a statistical outlier. In many fields, |z| > 1.96 corresponds to statistical significance at the 5% level.

How do I convert a z-score to a percentile?

The percentile (cumulative probability) corresponds to the area under the standard normal curve to the left of your z-score. For z = 1.0, the percentile is 84.13% — meaning 84.13% of values fall below this point. This calculator computes this conversion automatically using the standard normal CDF approximation.

What is the difference between z-score and t-score?

Both measure distance from the mean in standard deviation units, but t-scores are used when the population standard deviation is unknown and estimated from a sample (which is most of the time in practice). T-scores follow a t-distribution rather than normal distribution, and the shape depends on sample size (degrees of freedom). Z-scores assume you know the true population σ.

Can z-scores be negative?

Yes. A negative z-score simply means the value is below the mean. A z-score of −1.5 means the value is 1.5 standard deviations below average. There's no limit to how negative or positive a z-score can be, though values beyond ±4 are extremely rare in practice.

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How Far Is This Value From Average? Z-Score Calculator

How far is this value from average?

What This Does

A z-score answers one of the most fundamental questions in statistics: relative to the average, how unusual is this particular value? Z-scores express distance from the mean in units of standard deviations — making them the universal currency for comparing measurements across completely different scales. A student who scored 85 on a test with mean 75 and standard deviation 8 has a z-score of 1.25 — meaning their score is 1.25 standard deviations above average. That z-score can be looked up in a standard normal table (or calculated directly) to find that roughly 89.4% of students scored lower. The same z-score framework applies to any normally distributed data: heights, IQ scores, manufacturing tolerances, investment returns, blood pressure readings, and standardized test scores. Z-scores are the foundation of statistical inference. They power confidence intervals, hypothesis tests, quality control limits, and the calculation of p-values. Understanding z-scores means understanding how to answer: "Is this result surprising, or within normal variation?" — which is the central question of statistics. This calculator computes the z-score from any value, mean, and standard deviation, then converts it to a percentile (cumulative probability) and one-tailed probability — answering how common or rare the observation is.

When Should You Use This?

→Finding how many standard deviations a test score, measurement, or result is from the mean
→Converting a raw score to a percentile rank
→Checking whether a value is within normal range or an outlier
→Comparing scores from different tests or scales on equal footing
→Statistics homework involving normal distribution, hypothesis testing, or confidence intervals

Example Scenario

A pediatrician measures a 6-year-old boy's height as 46.5 inches. The population mean for 6-year-old boys is 45.5 inches with SD = 2.1 inches. Z-score = (46.5 − 45.5) / 2.1 = 0.476. Percentile: 68th percentile — this child is taller than 68% of boys his age. The doctor notes the child is in a completely normal range (z-score between −2 and +2 covers 95% of children).

Z-Score Calculator

Percentile, Probability & Normal Distribution

Enter any value, mean, and standard deviation — get z-score, percentile, and full probability breakdown in real time.

Value (x)

Mean (μ)

Std Deviation (σ)

Quick examples

About This Calculator

This z-score calculator converts any value to a z-score (standard score), percentile, and one-tailed and two-tailed probability in real time. Enter the value, mean, and standard deviation and all results update instantly. The formula z = (x − μ) / σ is applied exactly, and the percentile is derived from the standard normal cumulative distribution function Φ(z) using the Abramowitz & Stegun approximation, accurate to within 1.5 × 10⁻⁷.

The Bell Curve tab renders an interactive normal distribution area chart with the region below your value shaded in your tier colour, making the percentile immediately visual. The shaded area represents P(X ≤ x) — the cumulative probability to the left of your z-score. Dashed vertical reference lines mark ±1σ and ±2σ boundaries. The Landmarks tab provides a bar chart of standard z-score reference points (−3 to +3) with your value highlighted alongside a full probability table.

The Context tab places your z-score on a horizontal bar chart alongside six real-world benchmarks — SAT scores, IQ, height, blood pressure, stock returns, and employee performance ratings — giving immediate intuition for what your score means in practice. Quick example presets let you load common scenarios (test score, IQ 130, SAT 1400, average, outlier) with a single click. Use Export Full Report to generate a print-ready PDF with all probability data, the landmark table, and key insights.

Results are for informational purposes only.