Normal Distribution Calculator
Calculate cumulative probabilities and Z-scores for standard and custom Gaussian normal curves.
Standard Score (Z)
1.0000
1.0 standard deviations above mean
Percentile Rank
84.13%
Percentage of population below X
Left-tail \(P(X < 115)\):
0.84134
Right-tail \(P(X > 115)\):
0.15866
Normal Distribution Formulas & How to Calculate
View All Math Formulas Handbook →Probability Density Function (PDF)
Equation of the Gaussian bell curve:
$$ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} $$
Standard normal ($\mu = 0, \sigma = 1$):
$$ \phi(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}} $$
Standard Score (Z-Score) Formula
Number of standard deviations $x$ is from the mean:
$$ z = \frac{x - \mu}{\sigma} $$
How to calculate:
$$ \text{Step 1: } \text{Deviation} = x - \mu, \quad \text{Step 2: } z = \frac{\text{Deviation}}{\sigma} $$
Cumulative Distribution Function (CDF)
Left-tail probability representing the area under the bell curve:
$$ P(X \le x) = \Phi(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z} e^{-\frac{t^2}{2}} \, dt = \frac{1}{2} \left[1 + \operatorname{erf}\left(\frac{z}{\sqrt{2}}\right)\right] $$
Right-tail (greater than $x$) probability:
$$ P(X > x) = 1 - P(X \le x) = 1 - \Phi(z) $$