Question

Which measure will be affected by an outlier the most?

a) mean
b) median
c) range
d) mode

Answer 1

Range

Explanation:

An outlier is a data point that is distant from the other observations. For instance, in a data set of `{1,2,2,3,26}`, 26 is an outlier. There is a formula to determine the range of what isn't an outlier, but just because a number doesn't fall in that range doesnt necessarily make it an outlier, as there may be other factors to consider.

The `color(red)(median)` is the middle number of a set of numerically ordered numbers. If the number of values in the set is odd, then the `color(red)(median)` is the central number, with equal amounts of data on both its left and its right. If the set has an even number of values, then the `color(red)(median)` is the average of the two central numbers. For example, in the set of `{1,2,3,4,5,6,7,8}`, there is an even amount of numbers, therefore we must find the mean of the two central numbers, which results in
`(5+4)/2=4.5`, the `color(red)(median)` .

The `color(green)("range")` `r` is the distance from the highest value to the lowest value, and is calculated as `r=h-l`, where `h` is the highest value, and `l` is the lowest value. So if we have a set of `{52,54,56,58,60}`, we get `r=60-52=8`, so the `color(green)("range")` is 8.

Given what we now know, it is correct to say that an outlier will affect the `color(green)(ran)` `color(green)(g)` `color(green)(e)` the most. This is because the `color(red)(median)` is always in the centre of the data and the `color(green)(ran)` `color(green)(g)` `color(green)(e)` is always at the ends of the data, and since the outlier is always an extreme, it will always be closer to the `color(green)(ran)` `color(green)(g)` `color(green)(e)` then the `color(red)(median)`.

For example, take the set `{1,2,3,4,100}`, with 100 as the outlier. The `color(green)(ran)` `color(green)(g)` `color(green)(e)` of this set is `r=100-1=99`, while the `color(red)(median)` is 3. If we take the outlier 100 out, so the set is now `{1,2,3,4}`, the `color(green)(ran)` `color(green)(g)` `color(green)(e)` becomes `4-1=3`, while the `color(red)(median)` becomes `(3+2)/2=2.5`. Evidently, it was the `color(green)(ran)` `color(green)(g)` `color(green)(e)` which was affected the most.

https://mathspace.co/learn/world-of-maths/univariate-data/effects-of-outliers-12017/things-out-of-the-norm-601/

I hope I helped!