This free Normal Distribution Calculator lets you calculate probabilities, percentiles, and z-scores for the normal (Gaussian) distribution. Enter the mean (μ) and standard deviation (σ) to instantly find the PDF, CDF, right-tail probability, interval probability, or quantiles with step-by-step explanations.
The normal distribution (also called Gaussian distribution or bell curve) is the most widely used probability distribution in statistics.
$$ f(x) = \frac{1}{\sigma \sqrt{2\pi}} \; e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$
$$ F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) \, dt $$
$$ z = \frac{x - \mu}{\sigma} $$
Enter the mean (μ) and standard deviation (σ) of your distribution.
Choose your mode:
Point → Find PDF, CDF, and right-tail probability at a given x.
Interval → Calculate P(a ≤ X ≤ b) between two bounds.
Quantile → Find the value of x for a given cumulative probability (p).
Click Calculate to see results and detailed steps.
Copy or reset results as needed.
Normal(0, 1), x = 1.25
$$ z = \frac{1.25 - 0}{1} = 1.25 $$ $$ \Phi(1.25) = 0.8944 \;\;\Rightarrow\;\; P(X > 1.25) = 1 - 0.8944 = 0.1056 $$
Normal(0, 1), a = -1, b = 2
$$ P(-1 \leq X \leq 2) = \Phi(2) - \Phi(-1) = 0.9772 - 0.1587 = 0.8185 $$
Normal(0, 1), p = 0.975
$$ z = \Phi^{-1}(0.975) = 1.96 \;\;\Rightarrow\;\; x = 0 + 1 \times 1.96 = 1.96 $$
Statistics: hypothesis testing, confidence intervals.
Finance: stock returns, risk models.
Psychology & Education: test scores, IQ.
Manufacturing & Quality Control: process variation.
Natural & Social Sciences: measurement errors, population studies.
Q: How do I know if my data is normally distributed?
A: Use a histogram, Q-Q plot, or statistical tests like Shapiro–Wilk.
Q: Can probabilities be negative in the normal distribution?
A: No, probabilities range between 0 and 1, but the PDF (density) can be greater than 1 if σ < 1.
Q: What’s the difference between PDF and CDF?
A: PDF gives relative likelihood at a point, while CDF gives cumulative probability up to that point.
Q: Why use z-scores?
A: Z-scores standardize data, allowing comparisons across different normal distributions.