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How to use
A normal distribution is a continuous distribution defined by mean μ and standard deviation σ. The calculator standardizes values to z-scores, evaluates cumulative probability, and can invert a percentile back to x.
- Enter the mean and a positive standard deviation.
- Choose left-tail, right-tail, between-values, or inverse-percentile mode.
- Enter x, interval bounds, or the desired percentile.
- Review the z-score and probability in decimal and percent form.
How it is calculated
Example
Example: with mean μ=100 and standard deviation σ=15, the bounds 70.6 and 129.4 become z=(70.6−100)÷15=−1.96 and z=(129.4−100)÷15=1.96. Between-values mode gives a probability of about 95%. In inverse mode, the 97.5th percentile gives x≈100+1.96×15=129.4.
Important notes
The browser calculates numerical approximations. Extremely small tail probabilities may be displayed as 0% or 100% after approximation or rounding. A calculated probability does not establish that your real data follow a normal distribution; check whether this model fits before interpreting it.
Worked examples and interpreting results
In a normal distribution with mean 100 and standard deviation 15, 50% lies below the mean. About 68.269% lies between 85 and 115, within one standard deviation of the mean.
| Case | Calculation | Result |
|---|---|---|
| Below the mean | Φ(0) × 100 | 50 % |
| Within one standard deviation | (Φ(1) − Φ(−1)) × 100 | 68.268949 % |
How to check the result
For the first result select left probability and x=100; for the interval select between and enter 85 and 115. Probabilities depend on the normal model, not automatically on arbitrary data.
Frequently asked questions
What is a z-score?
It is the number of standard deviations x lies from the mean: z=(x−μ)/σ.
How is a between probability computed?
The lower cumulative probability is subtracted from the upper cumulative probability.
Can I use μ=0 and σ=1?
Yes. That is the standard normal distribution.