Normal Distribution Calculator

The normal distribution is the symmetric bell-shaped curve that describes many measurements, such as heights and test scores. This calculator finds the probability that a value falls below, above or between given values. It converts to z-scores, uses the standard normal distribution, and shades the matching area under the curve.

Try:

How to use the normal distribution calculator

  1. Enter the mean μ and standard deviation σ.
  2. Choose below, above or between, and enter the value(s).
  3. Press Calculate to see the probability and the shaded curve.

Formula

z=x−μσz = \frac{x - \mu}{\sigma}
f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Worked example: IQ between 85 and 115

  • Mean μ: 100
  • Standard deviation σ: 15
  • Find: P(a < X < b)
  • a: 85
  • b: 115

✓ Answer checked

P(85<X<115)≈0.68268949P(85 < X < 115) \approx 0.68268949
As a percentage
68.2689%
z-score(s)
za=−1,  zb=1z_a = -1,\; z_b = 1
4060801001201401600.0050.010.0150.020.0250.03
The shaded area under the bell curve is the probability, 68.269%.
Step-by-step working (2 steps)

  1. Standardise z = (x − μ)/σ

    Convert each value to a z-score: how many standard deviations it is from the mean.

    za=85−10015=−1,zb=115−10015=1z_a = \frac{85 - 100}{15} = -1,\quad z_b = \frac{115 - 100}{15} = 1

  2. Use the standard normal CDF Φ Standard normal distribution

    The area between is Φ(z_b) − Φ(z_a).

    Φ(1)−Φ(−1)≈0.68268949\Phi(1) - \Phi(-1) \approx 0.68268949
Formulas used
z=x−μσz = \frac{x - \mu}{\sigma}
f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
How this was checked
  • ✓ Numerically integrating the bell curve gives 0.68268949, the same probability.

Probabilities rounded to 8 significant figures.

Frequently asked questions

What percentage is within one standard deviation?

About 68.27% of values lie within one standard deviation of the mean, about 95.45% within two, and about 99.73% within three.

What is a z-score?

The number of standard deviations a value is from the mean: z = (x − μ) ÷ σ.

How accurate are the probabilities?

They are calculated to around 12 decimal places and checked by numerically integrating the bell curve, then shown to 8 significant figures.

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