SOLUTION: The mean of a set of numbers is 500. a. If 10 is added to each of the numbers in the set, then what will be the mean of the new set? b. If each of the numbers in the set are mult

Question 1190761: The mean of a set of numbers is 500.
a. If 10 is added to each of the numbers in the set, then what will be the mean of the new set?
b. If each of the numbers in the set are multiplied by -5, then what will be the mean of the new
set?
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

Let's say we had the set
{400,500,600}
The mean is 500. We can check as such
(400+500+600)/3 = 1500/3 = 500

Now let's add 10 to each number
{410,510,610}
and repeat the process of finding the mean
(410+510+610)/3 = 1530/3 = 510
The mean has increased by 10.

We can rewrite that equation like so
(410+510+610)/3 = (400+500+600 + 10+10+10)/3
(410+510+610)/3 = 500 + (10+10+10)/3
(410+510+610)/3 = 500 + (3*10)/3
(410+510+610)/3 = 500 + 10
(410+510+610)/3 = 510
If we add 10 to each number in the set, then the mean goes up by 10.

This can be applied in a more general sense, and it doesn't only work with this particular set (nor does it only apply to sets of 3 items)

{x1,x2,x3,...,xn} = original set of values
S = sum of the original items = x1+x2+x3+...+xn
S/n = original mean

Let's say we add on k to each number
This is equivalent to adding nk because there are n copies of k being added on.
So,
new mean = (S+nk)/n
new mean = (S/n)+(nk/n)
new mean = (old mean)+k
Therefore, what happened with the "adding 10" example earlier wasn't just a coincidence.

----------------------

Now let's consider multiplying each item in the set {x1,x2,...,xn} by some constant R
We would get this new set {Rx1,Rx2,...,Rxn}

Let's find the new mean of this scaled set.
new mean = (sum of the x values)/n
new mean = (Rx1+Rx2+...+Rxn)/n
new mean = (R*(x1+x2+...+xn))/n
new mean = (R*S)/n
new mean = R*(S/n)
new mean = R*(old mean)

So if we were to multiply each value by R = -5, then,
new mean = R*(old mean)
new mean = -5*(500)
new mean = -2500

Or you could go back to the example with 3 items from earlier
{400,500,600}
Multiply everything by -5 to get
{-2000,-2500,-3000}
Then compute the mean
(-2000+(-2500)+(-3000))/3 = -7500/3 = -2500

To summarize:
If you add 10 to each number, then the mean goes up by 10
If you multiply each number by -5, then the mean is multiplied by -5

============================================
Answers:
a) 510
b) -2500

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