Here

and

is a gamma random variable with shape parameter

and scale parameter

. The mean and variance of

are

and

. So variance stabilization theory suggests that the logarithmic transformation should be used. This Demonstration shows that the value of

that works best for making the distribution of

symmetric depends on the shape

. The case of

with

shown in the thumbnail works fairly well. With

, try experimenting with other shape parameters

to see how well this transformation preserves symmetry. Compare how well the logarithmic transformation,

, works to make the distribution symmetric. With

fixed, explore the shapes of various gamma distributions. Note that as

increases, the gamma distribution approximates the normal distribution. Changing the scale parameter

enlarges or shrinks the distribution but has no effect on the skewness or lack of symmetry.