Lesson Transformations in Statistics (adding the same value to each item in a list)

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n = some positive integer
Let X = {

} be a list of n values. Repeated items are possible.

Define m%5B1%5D

to be the arithmetic mean of list X
We'll add up all the items in list X, then divide by n which is the number of values in this list.

Now add some value to each item.
I'll add k to each value
k is any real number

X+k = {

}
Let

be the arithmetic mean of list X+k

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Claim: m%5B2%5D+=+m%5B1%5D%2Bk

Proof:

This is a fairly straightforward computation.

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A numeric example

X = {1,2,3,4,10}
m%5B1%5D+=+%281%2B2%2B3%2B4%2B10%29%2F5

Let's add k = 3 to each item, and compute the new mean.

X+3 = {1+3,2+3,3+3,4+3,10+3} = {4,5,6,7,13}
m%5B2%5D+=+%284%2B5%2B6%2B7%2B13%29%2F5

Note that

which is exactly the same as the k value we added to each item
In other words m%5B2%5D-m%5B1%5D+=+3

rearranges to m%5B2%5D+=+m%5B1%5D%2B3

which matches the form m%5B2%5D+=+m%5B1%5D%2Bk

This shows that if we add 3 to each item in the original list, then the mean has also increased by 3.
Replace '3' with any other number you like to try out another example.

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