Gaussian Distribution Entropy Calculator

Calculator

Input method

Mean μ

Standard deviation σ

Variance σ²

Dimension k

Covariance determinant |Σ|

Entropy base

Lower interval limit

Upper interval limit

Sample variance estimator

Sample data

Notes

Use the selected input method. Keep units consistent.

Example data table

Case	Dimension	Variance	Determinant	Entropy in nats	Use case
Unit Gaussian	1	1	1	1.418939	Reference noise model
Wide Gaussian	1	4	4	2.112086	Large uncertainty
Three variable model	3	4 each	64	6.336257	Vector physics data

Formula used

For one Gaussian variable, the calculator uses:

h(X) = 1/2 ln(2πeσ²)

For a multivariate Gaussian vector with dimension k, it uses:

h(X) = 1/2 ln((2πe)^k|Σ|)

Here, σ² is variance, Σ is the covariance matrix, and |Σ| is its determinant.

Base conversion is done by dividing nats by ln(2) for bits or ln(10) for bans.

Entropy power is calculated as N = exp(2h/k) / (2πe).

How to use this calculator

Select the input method that matches your available data.
Enter standard deviation, variance, covariance determinant, or sample values.
Set dimension one for a single Gaussian variable.
Increase dimension for a multivariate Gaussian vector.
Choose nats, bits, or bans for the reported entropy.
Use interval limits only for a one dimensional probability check.
Press Calculate to view entropy above the form.
Use CSV or PDF buttons to save your result.

Gaussian entropy in physics

Gaussian entropy tells how spread out a normal distribution is. In physics, it helps describe uncertainty in position, velocity, noise, heat motion, and model error. A narrow distribution has lower differential entropy. A wide distribution has higher differential entropy. The mean can shift the curve, yet it does not change entropy. The variance controls the spread, so it controls the final value.

Why variance matters

For one Gaussian variable, entropy depends on variance through a logarithm. This is useful because doubling spread does not double entropy directly. The calculator can use standard deviation, variance, or estimated sample data. It can also handle a multivariate case by using dimension and covariance determinant. That option is helpful when several related physical variables are studied together.

Physics meaning

Entropy here is differential entropy, not a simple count of states. It can be negative when units or very small variance make the density highly concentrated. That is normal for continuous variables. The value becomes more meaningful when the scale and units are consistent. In thermal physics, Gaussian models often appear in Brownian motion, sensor noise, molecular velocity components, and uncertainty propagation.

Advanced interpretation

The calculator also reports entropy power. Entropy power converts entropy into an equivalent Gaussian variance measure. For a one dimensional Gaussian, it matches the variance. For higher dimensions, it gives a compact spread measure per dimension. Effective volume, calculated as the exponential of entropy in nats, gives another view of the distribution size.

Practical use

Use this tool when you know the standard deviation or variance of a modeled Gaussian process. Enter dimension one for a single variable. Increase dimension for vector data. Use covariance determinant when variables are correlated. Choose bits for information work, nats for theory, or bans for base ten reporting.

Checks and cautions

The determinant must be positive for a valid covariance matrix. Standard deviation and variance must also be positive. Sample estimates depend on data quality. Results should be interpreted with physical units, instrument scale, and modeling assumptions in mind. Reliable reporting also needs a clear base. Changing from nats to bits only rescales the same entropy. It does not change the distribution itself or its physical spread during analysis or review.

FAQs

What is Gaussian distribution entropy?

It is differential entropy for a normal distribution. It measures continuous uncertainty. Higher variance gives higher entropy, while the mean does not change it.

Can Gaussian entropy be negative?

Yes. Differential entropy can be negative for narrow continuous distributions. This does not mean probability is negative. It reflects density scale and units.

Does the mean affect Gaussian entropy?

No. Moving the center of a Gaussian curve does not change its spread. Entropy depends on variance or covariance, not location.

What is the covariance determinant?

It is the determinant of the covariance matrix. In multivariate Gaussian entropy, it represents joint spread across all dimensions.

When should I use bits?

Use bits when comparing entropy with information measures. Bits are calculated by dividing the value in nats by ln(2).

What is entropy power?

Entropy power converts entropy into an equivalent Gaussian variance measure. It is helpful for comparing spread across dimensions.

Can I use sample data?

Yes. Choose the sample input method and paste numbers separated by commas, spaces, or lines. The calculator estimates variance first.

Is this the same as thermodynamic entropy?

Not exactly. This is differential entropy for a probability density. It can support physics modeling, but it is not direct heat entropy.