A z-score says how many standard deviations a value sits above or below the mean. This computes it, and the percentile that corresponds to it under a normal (bell-shaped) distribution.
How it works
Z-score
z = (x − μ) / σ
x is the value, μ is the mean, σ is the standard deviation. The percentile is computed from the standard normal distribution’s cumulative probability, using a published numerical approximation with a documented maximum error of about 0.00002 percentage points — accurate for any practical purpose, but an approximation rather than an exact closed-form value, since no elementary closed form for this exists.
What a z-score does and does not tell you
A z-score of 2 means the value sits two standard deviations above the mean — nothing more, on its own. It says nothing about the shape of the underlying distribution unless you also assume that distribution is normal, which is why the percentile figure here is explicitly labelled as applying to a normal distribution. A z-score of 2 in a heavily skewed distribution does not correspond to the same percentile a normal distribution would give it.
How to use this calculator
- Enter the value, the mean and the standard deviation.
- Read the z-score and, if a normal distribution is a reasonable assumption for your data, the percentile.
Frequently asked questions
What does a negative z-score mean?
The value sits below the mean — a z-score of −1.5 is one and a half standard deviations below average, corresponding to a lower percentile than 50%.
Why is the percentile only an approximation?
Because the normal distribution’s cumulative probability has no exact elementary closed-form expression — every practical implementation, including this one, uses a numerical approximation. The one used here (Abramowitz & Stegun 7.1.26) has a published maximum error of about 1.5×10⁻⁷, far smaller than matters for any real use.
Does this work if my data isn’t normally distributed?
The z-score itself (how many standard deviations from the mean) is always valid. The percentile figure specifically assumes a normal distribution, and will be inaccurate — sometimes substantially — if your actual data is skewed or has a different shape.
Why can’t standard deviation be zero here?
A standard deviation of zero means every value in the distribution is identical, which makes “how many standard deviations away” undefined for anything except that one identical value — division by zero, not a meaningful score.
What’s a “typical” range for z-scores?
Most values in a roughly normal distribution fall between z = −2 and z = 2 (about 95% of them), with values beyond ±3 being genuinely unusual — about 0.3% of a normal distribution falls outside that range.
