Question

The standard deviation of a set of numbers is 22.2. What would the standard deviation be if each one of the numbers in the set was decreased by 4?

Answer 1

There would be no change in the standard deviation if each one of the numbers in the set was decreased by 4.

Explanation:

We are given that the standard deviation of a set of numbers is 22.2.

And we have to check that what would be the standard deviation, if each one of the numbers in the set was decreased by 4.

For concluding this, we will take a simple example;

Suppose that there are three set of numbers : 6, 8 and 10

Firstly, we will calculate the standard deviation for these numbers;

X
`X - \bar X`
`(X-\bar X)^(2)`

6 6 - 8 = -2 4

8 8 - 8 = 0 0

10 10 - 8 = 2 4

8

Here, the Mean of the three numbers is given by;

Mean,
`\bar X` =
`(6+8+10)/(3)` = 8

Now, standard deviation formula is given by;

Standard deviation, s =
`\sqrt{((\sum X - \bar X)^(2) )/(n-1) }`

=
`\sqrt{(8 )/(3-1) }` = 2

Now, we will decrease each of the three numbers by 4, we get ;

X
`X - \bar X`
`(X-\bar X)^(2)`

2 2 - 4 = -2 4

4 4 - 4 = 0 0

6 6 - 4 = 2 4

8

Here, the Mean of the three numbers is given by;

Mean,
`\bar X` =
`(2+4+6)/(3)` = 4

Now, standard deviation formula is given by;

Standard deviation, s =
`\sqrt{((\sum X - \bar X)^(2) )/(n-1) }`

=
`\sqrt{(8 )/(3-1) }` = 2

From this, we observe that even after decreasing each number by 4, the standard deviation remains same.

SO, we can conclude that if each one of the numbers in the set was decreased by 4 , the standard deviation of a set of numbers will be 22.2 only.