Statistics Definitions > What is a Scale Parameter?

Scale parameters played an important role in the development of probability and statistics. They are used with location parameters to determine the shape and location of a distribution. A scale parameter stretches or squeezes a graph.

### 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.

**Need help with a homework or test question?** With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. If you'd rather get 1:1 study help, Chegg Tutors offers 30 minutes of **free tutoring** to new users, so you can try them out before committing to a subscription.

If you prefer an **online interactive environment** to learn R and statistics, this *free R Tutorial by Datacamp* is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try *this Statistics with R track*.

**Comments? Need to post a correction?** Please post a comment on our *Facebook page*.

Check out our updated Privacy policy and Cookie Policy