Mean of the gamma distribution
For a two-parameter gamma distribution written with shape k and scale θ, the expected value is the product kθ.
Gamma distributions are also written with a rate parameter β = 1/θ. Under that convention the mean is α/β. Confirm the parameterization before entering values.
How to calculate a gamma mean
- Verify the convention: Confirm your second parameter is scale θ, not rate β.
- Enter shape: Provide a positive shape parameter.
- Enter scale: Provide a positive scale parameter in the units of the modeled variable.
- Calculate: The result is the theoretical population mean under the specified model.
Formula and variables
This calculator uses the shape-scale convention and assumes no separate location parameter.
E[X] = kθ- E[X] — Expected value
- Population mean of the modeled random variable (same as X)
- k — Shape
- Positive gamma shape parameter, also written α (dimensionless)
- θ — Scale
- Positive gamma scale parameter (same as X)
Shape two and scale three
A gamma model has k = 2 and θ = 3 hours.
- Shape
- 2
- Scale
- 3 hours
- E[X] = 2 × 3
Result: The mean is 6 hours.
This is a model expectation, not a claim that every observation is near six hours.
Understanding your results
Parameterization controls the result
The same numeric second parameter produces a different mean when interpreted as a rate.
- Shape-scale: mean = kθ.
- Shape-rate: mean = α/β.
- A location-shifted gamma distribution adds the location parameter to the mean.
Assumptions
- Both parameters are strictly positive.
- The shape-scale, zero-location gamma parameterization is intended.
- The entered scale carries the units of the random variable.
Limitations
- Does not fit a distribution to observations.
- Does not calculate variance, quantiles, density, confidence intervals, or sample uncertainty.
- Does not include a location parameter.
Common mistakes
- Entering a rate as though it were a scale.
- Allowing zero or negative parameters.
- Confusing the theoretical mean with a sample average.
- Omitting the units carried by the scale parameter.
Practical use cases
Probability coursework
Check an expected-value calculation for a gamma model.
Model review
Translate fitted shape and scale values into an interpretable average.
Frequently asked questions
What is the mean with a rate parameter?
For shape α and rate β, the mean is α/β because scale θ equals 1/β.
Can gamma shape be less than one?
Yes. Shape must be positive, but it can be between zero and one.
What units does the answer use?
The answer has the same units as the scale parameter and the modeled random variable.
Sources and review
- Gamma Distribution Parameterizations — National Institute of Standards and Technology. Accessed 2026-07-13.
Reviewed 2026-07-13.