Question

If I subtract a constant B to all of the numbers, the measures of the center will ___ (go down by B / not change) and the measures of spread will ___ (go down by B / not change). If I divide all of the numbers in a list by a constant A, the measures of the center will ___ (be divided by A / not change) and the measures of spread will ___ (be divided by A / not change). is an example of a _____ ________. Another example is

Answer 1

Answer (Part 1):

If I subtract a constant B to all of the numbers, the measures of the center will go down by B and the measures of spread will not change.

Explanation:

Let the initial mean (measure of center) be and the standard deviation (measure of spread) be . In other words,

and

Now the new Expectation (mean) and Standard Deviation (SD) are:

(Mean of a constant is that number itself)

and

or,   (SD of a constant is zero)

Example:

Let the initial set of observations be: 7, 2, 3.

Here, E(X) = = (7+2+3)/3 = 4

and

(Sample SD Formula)

  

If we subtract 2 (=B) from all the numbers, then the new set of observations are: 5, 0, 1.

Now, E(Y) = (5+0+1)/3 = 2

Note that E(Y) = E(X) - 2 = E(X) - B.

Coming to the SD,

(Sample SD Formula)

  

  

We will get similar results for other measures of center and spread.

Now, consider the second part.

Answer (Part 2):

If I divide all of the numbers in a list by a constant A, the measures of the center will be divided by A and the measures of spread will be divided by A.

Explanation:

As earlier, initial mean be and initial standard deviation (SD) be .

i.e., and .

Now the new Expectation (mean) and Standard Deviation (SD) are:

  

and

Example(continued):

We consider the same example as in the previous part.

We have,

E(X) = 4 and

.

If we divide each number by 5(=A), then the new set of observations are: 1.4, 0.4, 0.6.

Now, E(Z) = (1.4+0.4+0.6)/3 = 0.8

Note that E(Y) = 4/5 = E(X)/A

Coming to the SD,

(Sample SD Formula)

  

  

  

Note that

We will get similar results for other measures of center and spread.