Gamma Distribution Formula: Shape, Scale, Rate and Worked Example

The gamma distribution is a continuous probability distribution for positive, right-skewed values such as waiting times, durations and amounts. Its density is f(x) = x^k-1 e^-x/θ / (θ^k Γ(k)) for x ≥ 0, where k is the shape and θ is the scale (the rate is 1/θ). Its mean is kθ and its variance is kθ². The most common source of error is mixing up scale and rate, so this page maps both forms onto Excel, R and Python.

What Is the Gamma Distribution?

The gamma distribution describes a non-negative continuous variable whose values pile up near the low end and trail off to the right. It has two parameters, a shape and a scale, and it includes the exponential and chi-square distributions as special cases. A standard use is the waiting time until the k-th event of a Poisson process (for example, the time until the fifth call arrives when calls come at a steady average rate). It is also the distribution behind the Gamma-Poisson mixture that produces the negative binomial distribution. For gamma-family priors, see Bayesian hierarchical models.

Gamma Distribution Formula (Shape, Scale, Rate)

The NIST/SEMATECH e-Handbook gives the general form with shape γ, location μ and scale β: f(x) = ((x-μ)/β)^γ-1 exp(-(x-μ)/β) / (β Γ(γ)), for x ≥ μ and γ, β > 0. The case μ = 0 and β = 1 is the standard gamma distribution, f(x) = x^γ-1 e^-x / Γ(γ). Most software uses the two-parameter form with μ = 0, written here with k for shape and θ for scale:

f(x) = x^k-1 e^-x/θ / (θ^k Γ(k)), x ≥ 0, k > 0, θ > 0

• x: the observed value (time, amount), which cannot be negative.
• k (shape): controls the form of the curve. Called alpha in Excel, a in SciPy and shape in R.
• θ (scale): stretches the curve along the x-axis, in the same units as x.
• λ = 1/θ (rate): the same information expressed per unit of x. The alternative density is λ^k x^k-1 e^-λx / Γ(k).
• Γ(k): the gamma function; Γ(k) = (k-1)! when k is a positive integer.

Quantity	Shape and scale	Shape and rate
Mean	kθ	k/λ
Variance	kθ2	k/λ2
Mode (k ≥ 1)	(k-1)θ	(k-1)/λ
Skewness	2/√k	2/√k

The R documentation states the mean as a×s and the variance as a×s² for shape a and scale s, and says that the scale defaults to 1/rate. SciPy states the same equivalence: its alpha-beta form is equivalent to the standard form with scale = 1/beta. The NIST skewness for the standard form is 2/√(shape), and the scale does not change it.

Gamma Distribution Examples: Worked Waiting-Time Example

This example is illustrative, with numbers chosen for easy arithmetic. Calls arrive at an average of 2 per minute, and we ask how long it takes until the fifth call. The waiting time T follows a gamma distribution with shape k = 5 and rate λ = 2 per minute, so the scale is θ = 1/2 = 0.5 minute.

1. Mean: kθ = 5 × 0.5 = 2.5 minutes (check with the rate form: 5/2 = 2.5).
2. Variance and SD: kθ² = 5 × 0.25 = 1.25 minutes² (rate form: 5/4 = 1.25), so SD = √1.25 = 1.118 minutes.
3. Mode: (k-1)θ = 4 × 0.5 = 2.0 minutes. The most likely single value sits below the mean, as expected for right skew.
4. Density at 2 minutes: f(2) = 2⁴ e^-4 / (0.5⁵ × 24) = 16 × 0.018316 / 0.75 = 0.3907 per minute.
5. Probability the fifth call has not arrived by 2 minutes: P(T > 2) = 0.6288, and P(T ≤ 2) = 0.3712. For an integer shape this equals the Poisson probability of at most 4 calls in 2 minutes (mean 4): e^-4(1 + 4 + 8 + 10.667 + 10.667) = 0.018316 × 34.333 = 0.6288.
6. 90th percentile: about 3.997 minutes, so nine times in ten the fifth call arrives within roughly 4 minutes.

Related cases: with k = 1 the same setup gives the exponential waiting time for the first call, and the count of calls in a fixed window is a Poisson distribution.

Interpretation

A gamma density value is not a probability. Probabilities come from areas under the curve (the cumulative distribution function), as in step 5 above.

What you see	How to read it
Shape k = 1	Exponential distribution (Microsoft documents that alpha = 1 returns the exponential).
Shape n/2 with scale 2 (n a positive integer)	Chi-square distribution with n degrees of freedom (Microsoft documents this relation for the cumulative form).
Positive integer shape	Also called the Erlang distribution (Microsoft).
Skewness 2/√k	k = 1 gives 2.0, k = 4 gives 1.0, k = 25 gives 0.4: the curve becomes more symmetric as shape grows. See left-skewed vs right-skewed data.
Mean greater than mode	For k > 1 the mean is kθ and the mode is (k-1)θ, so the mean is larger by exactly θ.

Assumptions and Common Mistakes

• Passing rate where scale is expected. Excel’s GAMMA.DIST takes beta as the scale, R’s dgamma takes rate or scale (scale = 1/rate), and SciPy takes scale. In the worked example, using 2 instead of 0.5 gives a mean of 10 minutes instead of 2.5.
• Negative values. The distribution is defined for x ≥ 0. Excel returns #NUM! for x < 0, alpha ≤ 0 or beta ≤ 0; R gives an error if you supply both scale and rate.
• Assuming it fits without checking. Compare a histogram of the data with the fitted density, or use a quantile-quantile plot, before relying on gamma probabilities.
• Zero values in the data. A strictly positive distribution cannot represent exact zeros without extra handling.
• Using it for counts. The gamma is continuous; counts belong to the Poisson or negative binomial families (see also quasi-Poisson vs negative binomial).

For a quick method-of-moments estimate, solve mean = kθ and variance = kθ² for the parameters: k = mean²/variance and θ = variance/mean.

How to Report It (APA Example)

State the distribution, both parameters with their units and parameterization, how they were estimated, and how fit was assessed. Illustrative wording:

Waiting times until the fifth call were modeled with a gamma distribution (shape = 5, scale = 0.5 min, equivalent to a rate of 2 per min), giving a mean of 2.5 min and an SD of 1.12 min. The probability that the fifth call had not arrived by 2 min was .63.

When parameters are estimated from data, also name the method (method of moments or maximum likelihood), report the software and version, and describe the goodness-of-fit check you used.

How to Calculate It in Excel, R or Python

Each snippet reproduces the worked example (shape 5, scale 0.5, rate 2).

Excel (beta is the scale; TRUE returns the cumulative probability, FALSE the density):

=GAMMA.DIST(2, 5, 0.5, TRUE)    -> 0.3712  (P(T <= 2))
=GAMMA.DIST(2, 5, 0.5, FALSE)   -> 0.3907  (density at 2)
=1 - GAMMA.DIST(2, 5, 0.5, TRUE) -> 0.6288  (P(T > 2))

R (rate and scale are alternatives; supply only one):

pgamma(2, shape = 5, rate = 2)         # 0.3712
dgamma(2, shape = 5, scale = 0.5)      # 0.3907
qgamma(0.9, shape = 5, scale = 0.5)    # 3.997

Python (SciPy stats; scale = 1/rate):

from scipy import stats
d = stats.gamma(a=5, scale=0.5)
d.cdf(2)    # 0.3712
d.pdf(2)    # 0.3907
d.ppf(0.9)  # 3.997
d.mean(), d.var()  # 2.5, 1.25

FAQ

What is gamma distribution?

It is a two-parameter continuous distribution for non-negative, right-skewed data. The shape and scale set its form, and it contains the exponential and chi-square distributions as special cases. It models the waiting time until the k-th event in a Poisson process.

What is the gamma distribution formula?

f(x) = x^k-1 e^-x/θ / (θ^k Γ(k)) for x ≥ 0, with mean kθ and variance kθ².

Is the second parameter scale or rate?

Both conventions exist. Rate is the reciprocal of scale. Excel and SciPy use scale; R accepts either rate or scale (scale defaults to 1/rate; supplying both is an error); many textbooks use rate.

What are some gamma distribution examples?

The waiting time until the fifth call in the example above is one; the same approach applies to any non-negative, right-skewed quantity once shape and scale are estimated; the worked example shows the full arithmetic.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods, Gamma Distribution: itl.nist.gov/div898/handbook/eda/section3/eda366b.htm (checked 2026-10-29).
• R Documentation, The Gamma Distribution (stats package): stat.ethz.ch/R-manual/R-devel/library/stats/html/GammaDist.html (checked 2026-10-29).
• SciPy documentation, scipy.stats.gamma: docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gamma.html (checked 2026-10-29).
• Microsoft Support, GAMMA.DIST function: support.microsoft.com/en-us/office/gamma-dist-function-9b6f1538-d11c-4d5f-8966-21f6a2201def (checked 2026-10-29).