Normal Distribution Calculator
Pick what you need, enter the mean and standard deviation, and get the probability (or the x value) together with the z-scores behind it.
Updated
Report a problem
Formula
- z-score:
z = (x − μ) ÷ σ - Below:
P(X < x) = Φ(z); above:P(X > x) = 1 − Φ(z) - Between:
P(a < X < b) = Φ(zb) − Φ(za) - Inverse:
x = μ + σ × Φ⁻¹(p)
Φ is the cumulative distribution function of the standard normal distribution.
Worked example
Scores with mean 100 and standard deviation 15:
- Below 115: z = 1, so P = 0.8413 (84.13%).
- Above 130: z = 2, so P = 0.0228.
- Between 85 and 130: 0.9772 − 0.1587 = 0.8186.
- The value with 95% below it: z = 1.6449, so x = 100 + 15 × 1.6449 = 124.67.
Frequently asked questions
How do I find the probability below a value?
Convert the value to a z-score, z = (x − μ) ÷ σ, and look up the standard normal CDF. With μ = 100, σ = 15 and x = 115, z = 1 and P(X < 115) = 0.8413, or 84.13%.
How do I find the probability between two values?
Subtract the two cumulative probabilities: P(a < X < b) = Φ(zb) − Φ(za). For μ = 100, σ = 15, between 85 and 130 that is 0.9772 − 0.1587 = 0.8186.
How do I find the x value for a given probability?
Use the inverse normal: x = μ + σ × z, where z is the point with that probability below it. The 95th percentile has z = 1.6449, so with μ = 100 and σ = 15, x = 124.67.
Is the probability of an exact value zero?
Yes. For a continuous distribution any single point has probability zero, so P(X < x) and P(X ≤ x) are the same. Only ranges have a non-zero probability.