Rescaling data is multiplying each member of a data set by a constant k; that is to say, transforming each number x to f(X), where f(x) = kx, and k and x are both real numbers. Rescaling will change the spread of your data as well as the position of your data points. What remains unchanged is the shape of distribution and the relative attributes of your curve.

## How does the mean, standard deviation, and variance change when a number is rescaled?

When data is rescaled the median, mean(μ), and standard deviation(σ) are all rescaled by the same constant. You will multiply by the scaling constant *k* to determine the new mean, median, or standard deviation.

- f(med)= k · med
- f(μ)= kμ
- f(σ) = kσ

The variance(σ^{2}) is rescaled by multiplying by the scaling constant squared.

- f(σ
^{2})=k^{2}σ^{2}

## Measures of Relative Standing

Note that percentile values and z-scores are not changed by rescaling. Since they are calculated as a ratio, the scaling constant cancels out.

## Examples of Rescaling Data

Scaling can be used if you need to change the type of your data measurements—say, from millimetres to kilometres or from acres to square inches. It is also often used to compare two data sets that are otherwise uncomparable because they use a different scale.

For instance, suppose teacher A grades her students on a 100 point basis, while another teacher B, teaching the same subject, grades her students on a basis of 170 points. Assume the difficulty level of all tests and homework are standardized between classes. To find out which class was doing better, you could rescale the class A data by mapping every member of data set x to f(x) where f(x)= 1.7x.

- A student with a grade of 99 would have his grade rescaled to 99 · 1.7= 153.
- A mean of 86 would translate to a rescaled mean of 86 · 1.7 = 146.2
- A standard deviation of 15 would be rescaled into a a standard deviation of 15 · 1.7 = 25.5
- If a student’s grade is in the 99th percentile pre-scaling, it will still be in the 99th percentile in the post-scaling results.

Sources:

https://www.cs.princeton.edu/courses/archive/fall08/cos436/Duda/PR_Mahal/scaling.htm

http://www.cs.xu.edu/math/math116/12f04/Ch06.pdf

https://www.stt.msu.edu/academics/classpages/uploads/SS15/200-4/Chapter6.pdf

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