Normal Distribution Calculator
Find probabilities and z-scores for any normal distribution.
What the Normal Distribution Calculator does
The normal distribution is the symmetric bell curve defined entirely by its mean and standard deviation. Probabilities come from the area under the curve, which this calculator evaluates via the cumulative distribution function.
Formula
z = (x − μ) ÷ σP(X ≤ x) = Φ(z), the standard normal CDFP(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a)
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| Mean (μ) | number | Yes | — |
| Standard deviation (σ) | number | Yes | Accepts more than 0. |
| Calculate | one of 4 options | Yes | — |
| x value | number | In some modes | Shown in 3 of the 4 modes. |
| Upper x value | number | In some modes | Shown Calculate is P(x₁ ≤ X ≤ x₂). |
| Probability | % | In some modes | Shown Calculate is Find x from a probability. |
How to use it
- Choose Calculate.
- Enter Mean (μ) and Standard deviation (σ).
- Fill in the remaining inputs the form shows for your choice.
- Select Calculate.
Worked example
IQ scores with a mean of 100 and standard deviation of 15 — what share exceed 130?
- Mean
- 100
- SD
- 15
- Mode
- P(X ≥ x)
- x
- 130
z = 2, so about 2.28% of people score above 130.
Reading the result
- "The empirical rule" table is a quick sanity check on any result here — if a computed probability for a ±1σ/2σ/3σ range doesn't roughly match 68%/95%/99.7%, something about the inputs is likely off.
Assumptions and limitations
- Every result assumes the data genuinely follows a normal distribution — the FAQ states many real distributions (income, wait times, returns) are skewed, so applying this to non-normal data gives a number with correct arithmetic but the wrong real-world meaning.
Common mistakes
- "Probability density at x" is a density, not a probability — it is not itself a probability (it can exceed 1 for a narrow distribution) and only becomes one when integrated over a range, which is what the CDF-based modes compute.
- Applying this calculator to a dataset without first checking it is approximately normal — the FAQ suggests plotting a histogram first, and skipping that risks a confidently wrong answer.
Frequently asked questions
What is the empirical rule?
For a normal distribution, roughly 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
Is my data normally distributed?
Many natural measurements approximate it, but income, wait times and financial returns are notably skewed. Plot a histogram before assuming normality.