Math & statistics
Normal Distribution Calculator
Enter a mean and standard deviation to find probabilities, z-scores and percentiles — with the curve shaded and every step shown. Works as a normalcdf and invNorm replacement.
P(X < a)
0.8413
—
z-score of a
—
z-score of b
—
percentile of a
Percentile → value (invNorm)
Enter a cumulative probability to get the cutoff value.
—
value x
—
z-score
Frequently asked questions
How do I convert a z-score to a percentile?
Look up the cumulative probability Φ(z) — the area under the standard normal curve to the left of z — and multiply by 100. For example z = 1.00 gives Φ(1.00) ≈ 0.8413, the 84th percentile. This calculator computes Φ(z) for you: set the mean to 0, the standard deviation to 1, and enter your z-score as the value.
How do I find the probability between two values?
Convert both values to z-scores, then subtract the cumulative probabilities: P(a < X < b) = Φ(zb) − Φ(za). Choose the “between” mode above and the calculator shows each step.
What is the 68–95–99.7 (empirical) rule?
For any normal distribution, about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. It is a quick approximation — the exact values are 68.27%, 95.45% and 99.73%.
What does invNorm do on a TI-84?
invNorm(p, μ, σ) returns the value x with cumulative probability p — the inverse of normalcdf. The “percentile → value” section below does the same thing: enter a probability and get the cutoff value and z-score.
What z-score corresponds to the 95th percentile?
z ≈ 1.645. Other common ones: 90th percentile ≈ 1.282, 97.5th ≈ 1.960, 99th ≈ 2.326.
What is the difference between normalcdf and invNorm?
normalcdf goes from values to a probability (area between two bounds). invNorm goes the other way — from a probability to the value that cuts off that area to its left.
Method: probabilities are computed from the standard normal CDF using a rational approximation of the complementary error function (accuracy ≈ 10⁻⁷) and Acklam’s algorithm for the inverse (accuracy ≈ 10⁻⁹) — the same math behind normalcdf and invNorm on a graphing calculator.