A shape parameter affects the general shape of a distribution, as the name implies; they are a family of distributions with various shapes. It also doesn't compress or shrink the graph (the job of the scale parameter). For certain distributions, it just defines the general shape of the graph.
The term "variance" has a specific meaning. The second central moment is always referred to as variance, and when we estimate or test the variance, we are estimating or testing this number. "Scale" is a broader term. It alludes to data dispersion in some form without committing to a discussion of the second central point.
Only the standard normal probability distribution has a scale equal to the standard deviation. The scale will not equal the standard deviation in most other types of distributions.
The SCP, also known as the scale with parameters, is a signal processing command that is frequently employed with analogue signals. The instruction will receive input, use the input minimum and maximum parameters, as well as the output min/max parameters, and scale the output accordingly.
When choosing a parametric model, it's best to pick one that produces the same type of new distribution when the random variable is multiplied by a constant. Scale distributions are a type of distribution.
