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Simple Useful Tools · Statistics

Normal Distribution Calculator

Calculate probabilities or find values in a normal distribution using a mean and standard deviation.

Parameters

Operation

Your Result

84.13%

Probability: 0.8413

Z-score: 1.0000

Normal curve

About 84.13% of values are expected to fall at or below X = 1.0000.

Calculation steps

1. Standardize the value:

z = (x − μ) / σ
z = (1.0000 − 0.0000) / 1.0000 = 1.0000

2. Calculate cumulative probability:

P(X ≤ 1.0000) = Φ(1.0000) = 0.8413

3. Percentage:

0.8413 × 100 = 84.13%

Summary

Mean
0.0000
Standard deviation
1.0000
X
1.0000
Z-score
1.0000
Probability
0.8413

What is a normal distribution?

A normal distribution is a continuous probability distribution that is symmetric around its mean. Values near the mean are the most common, and values that are farther from the mean become progressively less likely. Its graph forms the familiar bell curve. "Gaussian distribution" is another name for the same distribution.

The distribution is determined by two parameters:

  • μ — the mean
  • σ — the standard deviation

How to use the Normal Distribution Calculator

The calculator has two modes depending on what you want to find:

Find Probability

  • Less: Find the probability that a value is less than or equal to x, written as P(X ≤ x).
  • Greater: Find the probability that a value is greater than or equal to x, written as P(X ≥ x).
  • Between: Find the probability that a value falls between two boundaries, written as P(a ≤ X ≤ b).
  • Outside: Find the probability that a value falls outside two boundaries, written as P(X ≤ a or X ≥ b).

Find Value

  • Below: Find the value X for a cumulative probability at or below X.
  • Above: Find the value X for an upper-tail probability above X.

The Standard Normal preset automatically sets the mean to μ = 0 and the standard deviation to σ = 1.

Normal distribution probability and z-scores

To calculate probabilities for any normal distribution, we first standardize the value using the formula:

z = (x − μ) / σ
  • z = 0 means the value equals the mean.
  • A positive z means the value is above the mean.
  • A negative z means the value is below the mean.
  • The absolute value |z| measures the distance from the mean in standard deviations.

Once the value is standardized, probabilities are obtained from the standard normal cumulative distribution Φ(z).

The 68–95–99.7 rule

For approximately normally distributed data, this simple rule of thumb applies:

68%
within 1 standard deviation
95%
within 2 standard deviations
99.7%
within 3 standard deviations

Note: This is an approximation for normally distributed data. Not every real-world dataset is normal.

Worked example

Suppose we have a normal distribution with:

  • Mean μ = 70
  • Standard deviation σ = 10

We want to find the probability of a value being less than or equal to 85, which is P(X ≤ 85).

First, calculate the z-score:

z = (85 − 70) / 10 = 1.5

For P(X ≤ 85), the cumulative probability is approximately:

0.9332 (or 93.32%)

This means a value of 85 is 1.5 standard deviations above the mean, and about 93.32% of the distribution lies at or below 85.

Normal PDF vs. Normal CDF

PDF (Probability Density Function): describes the relative density or shape of the distribution at values of X.

CDF (Cumulative Distribution Function): gives the cumulative probability at or below X.

For a continuous distribution, probability comes from the area under the curve (using the CDF), not the height of the PDF at a single point.

Frequently asked questions

What is a normal distribution?
A normal distribution is a continuous probability distribution that is symmetric around its mean. Values near the mean are the most common, and as values get further from the mean in either direction, they become progressively less likely. Its graph is the familiar bell curve.
Is a Gaussian distribution the same as a normal distribution?
Yes. Gaussian distribution and normal distribution refer to the same family of probability distributions.
How do you calculate probability in a normal distribution?
To calculate probability in a normal distribution, you find the area under the bell curve for a specific range of values. This is typically done by converting the value to a standard normal z-score and looking up the corresponding cumulative probability, or by using a calculator like this one.
What is the 68–95–99.7 rule?
The 68-95-99.7 rule is an approximation for normally distributed data. It states that roughly 68% of the data falls within 1 standard deviation of the mean, 95% falls within 2 standard deviations, and 99.7% falls within 3 standard deviations.
Does a normal distribution always have a mean of 0 and standard deviation of 1?
No. Only the STANDARD normal distribution has a mean of 0 and a standard deviation of 1. Other normal distributions can have any mean and any positive standard deviation.
What is the difference between the normal PDF and CDF?
The Probability Density Function (PDF) describes the relative density or shape of the distribution at specific values. The Cumulative Distribution Function (CDF) gives the cumulative probability at or below a specific value. For continuous distributions, probability comes from the area under the curve (CDF), not the height of the PDF at a single point.
What does a z-score mean?
A z-score measures the distance of a value from the mean in terms of standard deviations. A z-score of 0 means the value equals the mean, a positive z-score is above the mean, and a negative z-score is below the mean.
How do I find a value from a probability or percentile?
Given a cumulative probability, use the inverse normal distribution to find the value corresponding to that probability. For an upper-tail probability, you first convert it to a left cumulative probability.