Area below z = 1
Z-score basis, Left, z = 1
P(Z ≤ 1) ≈ 0.84134475, or 84.13447461%
Boundary → probability
Normal Distribution Curve Calculator
Enter a value or z-score, choose an area, and calculate the probability with a clearly labeled shaded curve.
- Runs locally in your browser
- No sign-up
- Four shaded area modes
Ready example
Area below x = 1
Use the prefilled standard normal values, then choose Calculate normal probability.
Method
How normal curve probability works
Choose the values you already have
Use raw-value mode when a problem gives x together with a mean μ and standard deviation σ. Use z-score mode when the values have already been standardized. The z-score tells how many standard deviations a boundary lies above or below the mean.
Match the wording to an area
Left means the cumulative area below one boundary. Right means the area above it. Between selects the area inside two ordered boundaries, while Outside adds the two tails beyond those boundaries. The diagram and text both show which region is being calculated.
Standardize and evaluate the probability
For raw values, each boundary becomes z = (x − μ) / σ. The calculator then evaluates the normal cumulative distribution using stable tail and interval formulas. It reports the decimal probability, percentage, standardized boundary, steps and shaded curve together.
Checked examples
Worked normal curve examples
Area between two z-scores
Z-score basis, Between, −1 < Z < 1
P(−1 ≤ Z ≤ 1) ≈ 0.68268949, or 68.26894921%
Custom score interval
Raw basis, 85 < X < 115, μ = 100, σ = 15
The standardized bounds are −1 and 1, so the probability is ≈ 0.68268949.
FAQ
Normal distribution area questions
What area does a normal distribution calculator return?
It returns probability represented by area under the normal curve. The total area is 1. A left or right calculation uses one boundary; between and outside calculations use two ordered boundaries.
Should I enter a raw value or a z-score?
Enter a raw value when your number is on the original measurement scale and you know its mean and standard deviation. Enter a z-score when the problem already gives a standardized value. In raw mode this calculator shows the standardization step for you.
What is the difference between Between and Outside?
Between shades the single region inside the lower and upper boundaries. Outside shades both tails: values below the lower boundary plus values above the upper boundary. For continuous normal distributions, boundary inclusivity does not change the probability.
Why must standard deviation be greater than zero?
Standard deviation sets the distribution scale. A zero or negative value cannot define a normal distribution and would make z = (x − μ) / σ invalid, so the calculator rejects it before calculating.
Review and sources
Definitions and numerical references
Reviewed July 13, 2026
This educational calculator is designed for learning and routine checking. Verify medical, financial, legal, engineering-safety, or other high-stakes work independently.
- NIST normal distribution definition
- NIST standard normal cumulative distribution
- R normal distribution documentation
- Python NormalDist documentation
Have a percentile instead? Find the inverse normal boundary.