Question

The mean of a set of numbers is X. If each number is multiplied by z, then find the new mean.

Answer 1

Hint: In order to solve the question, first assume a set of numbers. Next, write their mean using the formula. Then, write the set of numbers when they are multiplied by z. Finally, write the new mean.

Formula Used:
${\rm{Mean}} = \dfrac{{{\rm{Sum of observations}}}}{{{\rm{Number of observations}}}}$

Complete step by step solution:
Given that
The mean of a set of numbers is X.
Suppose that the set of numbers is ${a_1},{a_2},{a_3},...,{a_n}$.
We know that
${\rm{Mean}} = \dfrac{{{\rm{Sum of observations}}}}{{{\rm{Number of observations}}}}$
That is
$X = \dfrac{{{a_1} + {a_2} + {a_3} + ... + {a_n}}}{n}$
Next, each number is multiplied by z. That is
The new set of numbers will be $z{a_1},z{a_2},z{a_3},...,z{a_n}$
So we can write that the new mean X’ as
$X' = \dfrac{{z{a_1} + z{a_2} + z{a_3} + ... + z{a_n}}}{n}$
$X' = \dfrac{{z\left( {{a_1} + {a_2} + {a_3} + ... + {a_n}} \right)}}{n}$
$X' = zX$

Hence the new mean is $zX$.
Additional Information
The mean of some numbers is the ratio of the sum of the numbers to the total number of numbers whose mean is found. Students should be aware that if they change the values of the numbers whose means are to be calculated, their mean will change as well.

Note: One can easily solve the question by assuming a set of numbers and then using the mean formula. Also, remember that both mean and arithmetic mean are the same. Remember that here the set of numbers is multiplied by z, so the mean can increase z times.