Calculate normal distribution

Find a z-score and lower, upper, or between-bound probability under a normal model.

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ToolNormal Distribution Calculator
Input
Output
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How it works

The observation is standardized as z = (x − mean) ÷ standard deviation. An Abramowitz–Stegun error-function approximation evaluates the cumulative probability, from which lower-tail, upper-tail, and between-bound results are assembled.

  • Mean zero and standard deviation one initially represent the standard normal.
  • Approximation error is roughly bounded by 1.5 × 10⁻⁷ rather than being symbolically exact.

Worked example

One standard deviation either side of the mean
Mean 100, standard deviation 15, band from 85 to 115
Input
											Mean: 100
Standard deviation: 15
Lower value: 85
Upper value: 115
Output
												Z score lower: -1
Z score upper: 1
Probability below lower: 0.1587
Probability above upper: 0.1587
Probability between: 0.6827

When to use this

Z-score exercises standardize observations, manufacturing models estimate tail probabilities, and statistics lessons compare probability between bounds.

Edge cases

  • A zero or negative standard deviation is rejected because it cannot define this distribution.
  • Very large absolute z values round to probabilities indistinguishable from zero or one at display precision.
  • Skewed, bounded, or heavy-tailed data can make a mathematically correct normal-tail result a poor model of observations.

References