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Calculation Machine

Normal Distribution Calculator

Probabilities under the bell curve.

Reviewed by the Calculation Machine team · How we check our maths

Your values

ⓘ Live results

Results
P(X < x)
0.908789
P(X > x)
0.091211
Z score
1.33333
Percentile
90.88%
P(x < X < x₂)
0.068461

Quick answer

The normal distribution converts any value to a z-score, then reads the area under the standard bell curve.

The formula

z = (x − μ) / σ

Worked example

Worked example

Using the values the calculator loads with.

Inputs

  • Mean μ100
  • Standard deviation σ15
  • Value x120
  • Second value (optional)130

Result

  • P(X < x)0.908789
  • P(X > x)0.091211
  • Z score1.33333
  • Percentile90.88%
  • P(x < X < x₂)0.068461

How to use the Normal Distribution Calculator

  1. 1Enter the population mean and standard deviation.
  2. 2Enter the value you are testing.
  3. 3Add a second value for a range probability.

Frequently asked questions

What is the 68-95-99.7 rule?

About 68% of data lies within 1 SD, 95% within 2 and 99.7% within 3.

Is my data normal?

Check a histogram or QQ plot; heavily skewed data needs different tools.

About this calculator