Fisher Information Calculator

Calculator Inputs

Use the responsive three, two, and one column layout for large, medium, and small screens.

Model

Success probability p

Choose a value inside the allowed parameter range.

Sample size n

Total information scales linearly with independent sample size.

Known standard deviation σ

Needed only for the normal mean model.

Example Data Table

Model	Parameter	n	I₁	Iₙ	CRLB
Bernoulli	p = 0.40	80	4.166667	333.333333	0.003000
Poisson	λ = 3.50	120	0.285714	34.285714	0.029167
Exponential	λ = 1.80	50	0.308642	15.432099	0.064800
Normal mean	σ = 2.40	64	0.173611	11.111111	0.090000
Normal variance	σ² = 1.60	40	0.195313	7.812500	0.128000
Normal standard deviation	σ = 1.60	40	0.781250	31.250000	0.032000

Formula Used

Fisher information measures how strongly an observable sample responds to a small parameter change. Larger values mean the data carries more precision about that parameter.

I(θ) = E[(∂/∂θ log f(X; θ))²]

I(θ) = −E[∂²/∂θ² log f(X; θ)]

Iₙ(θ) = n × I₁(θ)

Var(θ̂) ≥ 1 / Iₙ(θ)

Model	Per-observation Fisher information
Bernoulli(p)	I₁(p) = 1 / [p(1 − p)]
Poisson(λ)	I₁(λ) = 1 / λ
Exponential rate λ	I₁(λ) = 1 / λ²
Normal mean μ, known σ	I₁(μ) = 1 / σ²
Normal variance σ², known μ	I₁(σ²) = 1 / [2(σ²)²]
Normal standard deviation σ, known μ	I₁(σ) = 2 / σ²

How to Use This Calculator

Select the probability model matching your estimator problem.
Enter the parameter value for the chosen model.
Add the independent sample size n.
If you choose the normal mean model, provide the known standard deviation.
Press the calculate button to show information metrics above the form.
Review I₁, Iₙ, the Cramér–Rao lower bound, and minimum standard error.
Use the CSV or PDF buttons to export the displayed result summary.

Frequently Asked Questions

1. What does Fisher information measure?

It measures how much information an observable sample carries about an unknown parameter. Higher information means tighter achievable precision for unbiased estimators under regularity conditions.

2. Why does sample size increase Fisher information?

For independent observations, Fisher information adds across samples. Doubling the sample size doubles total information and reduces the minimum attainable standard error.

3. What is the difference between I₁ and Iₙ?

I₁ is information from one observation. Iₙ is total information from n independent observations, computed as n times the single-observation value.

4. What is the Cramér–Rao lower bound?

It is the minimum variance bound for unbiased estimators under standard assumptions. The bound equals the reciprocal of total Fisher information.

5. Why does the Bernoulli model explode near 0 or 1?

When p approaches the boundaries, small probability changes sharply alter the likelihood. That produces very large Fisher information values near 0 and 1.

6. Can I use this for observed Fisher information?

This page reports closed-form expected Fisher information for supported models. Observed information depends on the realized sample and requires a likelihood-based data calculation.

7. Does the normal mean model need the mean value?

The information for a normal mean with known standard deviation does not depend on the actual mean value. The calculator still stores your entered mean for reporting context.

8. When should I export the results?

Export when you need documentation for coursework, model comparison, review notes, or reporting. The CSV and PDF capture the current metrics shown in the result panel.