Normal distribution calculator

Enter the mean μ (average), standard deviation σ, and cutoff points, and this normal distribution calculator will compute the area (probability) under the normal distribution curve.

Enter the parameters of the normal distribution:

μ

σ

Above

Below

Between

and

Outside

and

Result:

μ = 3.1
σ = 0.4

Area (probability) = 0

The normal distribution, also known as the Gaussian distribution, is a symmetric, bell-shaped curve that describes the distribution of many natural phenomena. It is characterized by its mean (μ or m) and standard deviation (σ or SD), which determine its center and spread, respectively.

In a normal distribution, about 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

The normal distribution is widely used in statistics because of the Central Limit Theorem, which states that the sum of a large number of independent random variables tends to follow a normal distribution.

Many statistical tests, such as the z-test and t-test, assume that the data follows a normal distribution.

The curve is perfectly symmetrical, with the mean, median, and mode all located at the center of the distribution.

The tails of the normal distribution extend infinitely in both directions, but the probability of extreme values decreases rapidly.

Standardizing a normal distribution (subtracting the mean and dividing by the standard deviation) converts it into the standard normal distribution, which has a mean of 0 and a standard deviation of 1.

The normal distribution is often used in quality control, finance, and social sciences to model errors, returns, and other random variables.

While many real-world datasets approximate a normal distribution, deviations such as skewness or kurtosis can indicate that the data does not perfectly fit this model.

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