Enter the mean μ and standard deviation σ, then choose what to calculate and enter a score or probability.
Example: Heights are normally distributed with mean 170 cm and SD 10 cm — what fraction are shorter than 185 cm? Set μ = 170, σ = 10, choose P(X ≤ x), enter x = 185, click Calculate.
Hover over the chart to see the PDF — f(x) (probability density at x), the CDF — P(X ≤ x), and P(X > x).
Calculation steps
R Code
The following R code should produce the same results.
The Normal Distribution Calculator computes probabilities, quantiles, and density values for the normal distribution N(μ, σ²), where μ is the mean and σ is the standard deviation. It supports both the standard normal (Z) distribution and any normal distribution with custom μ and σ.
How to use: Enter the mean (μ) and standard deviation (σ), then enter either a score or a target probability, and click Calculate.
What you can calculate:
P(X ≤ 𝑥) — cumulative probability based on the score.
P(X ≥ 𝑥) — upper-tail probability (1 − CDF) based on the score.
P(𝑥₁ ≤ X ≤ 𝑥₂) — probability that X lies between 𝑥₁ and 𝑥₂.
P(X ≤ 𝑥₁) + P(X ≥ 𝑥₂) — probability that X lies outside the range 𝑥₁ to 𝑥₂.
PDF f(x) — value of the probability density function at 𝑥.
Score 𝑥 from P(X ≤ 𝑥) — find the score for a given cumulative probability (inverse CDF).
Score 𝑥 from P(X ≥ 𝑥) — find the score for a given upper-tail probability.
Scores 𝑥₁, 𝑥₂ — find both tail scores from two probabilities.
Z Score — the standard score, based on the mean and standard deviation.
The normal distribution calculator and z-score calculator use the normal distribution. The normal distribution is also known as the Gaussian distribution. The normal distribution is the most utilized in statistical analysis. This is because many natural processes follow the normal distribution, or resemble it For example, the height, weight, and measurement error follow the normal distribution.
The Normal distribution has a symmetrical "Bell Curve" shape. Most of the data centers around the average, and the frequency decreases as values move farther away from the center.
According to the Central Limit Theorem (CLT), the distribution of the sum or average of independent random variables tends to approach a normal distribution as the number of variables increases This means that the distribution of the average or sum of non-normal distributions may look like the normal distribution. The larger the sample size, the more the distribution resembles the normal distribution.
You may calculate values for any normal distribution, using the standard normal distribution. The standard normal distribution is a special case of the normal distribution. It contains the following parameters: a mean of 0 and a standard deviation of 1.
When X follows a normal distribution with mean (μ) and standard deviation (σ), the standardized value Z = (x-μ)/σ follows the standard normal distribution. As a result, values for any normal distribution can be calculated based on the standard normal distribution.
PDF(𝑥) = f(x) =
1
exp(-
(𝑥 - μ)2
)
σ√(2π)
2σ2
Z - Standard distribution score - normal distribution with μ=0 and σ=1.
Z =
𝑥 - μ
σ
Normal distribution PDF:
The Probability Density Function (PDF), indicated as f(x), is relevant to a continuous random variable. It explains the relative likelihood of a given value. It means that you would expect to encounter more values around a higher PDF than around a lower PDF. To calculate the probability that a random variable will fall in a specific range, you need to calculate the area under the density curve. You may do it by calculating the integral of the density. The probability of obtaining an exact value of a continuous random variable is zero.
