Calculator of Alexandria

Z-score and normal probability calculator

Calculate a z-score and its percentile under the normal curve, with the area above, below and between explained.

z-scorezz
Percentile (area below)
Area above
Area within ±z of the mean

Formula

z=x−μσz = \frac{x - \mu}{\sigma}
How it's derived

The formula

z=x−μσz = \frac{x - \mu}{\sigma}

xx is a value, μ\mu the mean and σ\sigma the standard deviation. The percentile is the area under the standard normal curve to the left of zz:

Φ(z)=12π∫−∞ze−t2/2 dt\Phi(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z} e^{-t^2/2}\, dt

Where it comes from

A z-score measures distance from the mean in units of standard deviations. Subtracting μ\mu moves the center to 0, and dividing by σ\sigma rescales the spread to 1. Any normal distribution then becomes the same standard normal curve, so one table, or one function, covers every case.

The bell curve itself comes from the central limit theorem. When a quantity is the sum of many small, independent effects, like measurement error or the height of a person, its distribution approaches this shape. That’s why it shows up everywhere.

The integral Φ(z)\Phi(z) has no formula in elementary functions. Statisticians used printed tables for decades. This calculator uses a standard polynomial approximation, accurate to about seven decimal places.

Worked example

SAT-style scores with mean 1050 and standard deviation 200. A score of 1350:

z=1350−1050200=1.5z = \frac{1350 - 1050}{200} = 1.5 Φ(1.5)=0.933\Phi(1.5) = 0.933

So 1350 is at about the 93rd percentile.

Rules of thumb

About 68% of values lie within 1 standard deviation of the mean, 95% within 2 (1.96 exactly), and 99.7% within 3.

Watch out for

  • Only for normal data. Incomes, waiting times and many biological measurements are skewed. Percentiles from this calculator would be wrong for them.
  • Sample vs population. With a small sample, use a t-distribution instead.

Sources

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