Statistics Definitions > What is a Scale Parameter?

A scale parameter stretches or squeezes a graph. They are used with location parameters to determine the shape and location of a distribution.

### Scale Parameter’s Effect on Graphs

The graph on the left has a scale parameter of 3; The graph on the right has a scale parameter of 1/3. The effect of changing the scale parameter from 3 to 1/3 is squeezing the graph: the graph on the left is between -10 and 10 on the horizontal axis while the graph on the right is between -2 and 2.

Scale parameters give **meaning **to graphs. In a standard normal model, the scale is equal to the standard deviation, σ. Without a scale on a graph, you can’t extract any information from it, even given that the area under the graph is 1. The top graph here is a standard normal distribution without any scale parameters. The bottom graph has scale parameters written as standard deviations:

The scale is equal to the standard deviation is ** only true for the standard normal probability distribution.** In most other distribution types, the scale will **not equal** the standard deviation.

### General Rules

The larger the scale parameter, the more spread out the distribution. The smaller the parameter, the more compressed the distribution. A scale parameter:

- …of
**zero**will result in a single, vertical line at 0 (a spike). - …between
**zero and 1**will squeeze the distribution horizontally. - …
**exactly 1**leaves the distribution unchanged. - …
**more than 1**stretches the graph horizontally.

The above graph shows the effect of parameters. As the graph shrinks horizontally with smaller scales, the graph also grows taller.* Why*? Because the area under the curve has to equal 1, so all the area has to go up if it’s width shrinks.

## References

Balakrishnan, N. and Cohen, A. C. Order Statistics and Inference. New York: Academic Press, 1991.

Wheelan, C. (2014). Naked Statistics: Stripping the Dread from the Data.