Normal Distribution Calculator
For a value on a normal distribution, get its z-score, the probability density at that point, and the probabilities of falling at or below and at or above it. Enter the value, the mean, and the standard deviation.
What this calculator does
The Normal Distribution Calculator estimates your result instantly from the values you enter, using standard statistics formulas. For a value on a normal distribution, get its z-score, the probability density at that point, and the probabilities of falling at or below and at or above it. Enter the value, the mean, and the standard deviation. Calculations run in your browser, so nothing you type is sent to a server, and the output is an estimate rather than a certified or official figure.
Result
About Normal Distribution Calculator
Enter a value x along with the distribution's mean μ and standard deviation σ, and this calculator standardizes it to a z-score, z = (x − μ) / σ. From z it evaluates the normal density f(x) and the cumulative probability using an error-function approximation of the standard normal CDF.
It reports four numbers: the z-score, the density at x, the probability P(X ≤ x), and its complement P(X ≥ x). The standard deviation must be greater than zero.
Formula & Mathematical Calculation
Z-Score = f(Value (x), Mean (μ), Standard deviation (σ))
Calculated according to standard stats formulas implemented in the calculator core.
Calculation Example
With value x = 70, mean μ = 50, and standard deviation σ = 10:
- Z-Score = (70 − 50) / 10 = 2
- Density f(x) ≈ 0.005399
- P(X ≤ x) ≈ 97.725%
- P(X ≥ x) ≈ 2.275%
So the value sits two standard deviations above the mean, with about 97.725% of the distribution at or below it.
Key Benefits
- Returns four results at once: the z-score, the density f(x), and both tail probabilities P(X<=x) and P(X>=x)
- Standardizes any value using your own mean and standard deviation, not just the standard normal
- Uses an error-function approximation for the cumulative probability, so no lookup table is needed
- Requires a positive standard deviation and shows results to several decimals; free and in-browser
How to Use this Calculator
-
Enter the value
Type the point of interest into Value (x). -
Enter the mean
Type the distribution's average into Mean, shown as the Greek letter mu. -
Enter the standard deviation
Type the spread into Standard deviation (sigma); it must be greater than 0. -
Calculate
Press Calculate to read the Z-Score, Density f(x), P(X<=x), and P(X>=x).
Assumptions & Limitations
What the calculation assumes
- Every value you enter is treated as an exact, known amount.
- Any rate or interval you enter is assumed to hold for the whole period unless the form asks for it separately.
- Results use standard published formulas and account only for the values entered above.
- Rounding is applied for display; the underlying calculation uses your full precision.
- Results are returned in the same units you enter, unless the tool states the output unit.
What it does not cover
- Normal Distribution Calculator produces a calculated result for reference and learning, not a certified or official measurement.
- Real-world outcomes differ when tolerances, rounding rules or local standards apply.
- Definitions and standards vary by region and change over time; verify the convention that applies to your case.
- Figures are only as accurate as the inputs you supply — check them before acting on the result.
- For work where the result carries real consequences, confirm it against an authoritative source.
Frequently Asked Questions
Free to use, no account needed. Calculations run in your browser, so nothing you type is sent to a server. Results are estimates — how we build and check our calculators.