Calculate the z-score and corresponding percentile from a value, mean and standard deviation.
The z-score measures how many standard deviations a value lies from the mean of a distribution. A z-score of 0 means the value equals the mean; +1 means it is one standard deviation above; −2 means two below. In a normal distribution, the z-score maps directly to a percentile: z = 0 is the 50th percentile, z = +1 is about the 84th, z = −1 is about the 16th, and z = ±2 covers roughly 95% of the data.
The formula is simple: z = (x − μ) / σ. The percentile is the area under the standard normal curve to the left of that z-score, computed here with an approximation that is accurate to five decimal places across the entire range. This approximation — the Abramowitz and Stegun formula — uses the complementary error function and avoids the need for a z-table, which is what this tool replaces.
Typical uses: checking whether a test score is above or below average relative to the class, determining how unusual a measurement is in a quality control process, and converting between raw scores and percentiles in educational testing. The z-score is meaningful only when the underlying distribution is approximately normal — for skewed or multimodal data, the percentile interpretation does not hold.
z = (x − μ) / σ. Percentile = Φ(z) × 100, where Φ is the cumulative distribution function of the standard normal distribution.
There is no universally good or bad z-score — it depends on the context. In many applications, values within ±2 are considered normal, and beyond ±3 are outliers.
Yes — a negative z-score means the value is below the mean. z = −1.5 means 1.5 standard deviations below average.
The z-score formula still works (it is just arithmetic), but the percentile interpretation assumes normality. For non-normal data, the actual percentile may differ significantly.
Yes — it computes the cumulative probability directly instead of looking it up in a table. The result is more precise than a table rounded to two decimal places.
No. The calculation runs in JavaScript on your device.
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