# The mean of 3 numbers is 7. The mean of 4 other numbers is 8. The mean of 5 other numbers is 11. What is the mean of all 12 numbers?

Aug 11, 2016

$9$.

#### Explanation:

Let ${\overline{x}}_{i}$ be the mean of a group ${G}_{i}$ of ${n}_{i}$ observations,

where, $i = 1 , 2 , 3 , \ldots , m$.

Then, the mean $\overline{x}$ of the combined group

$G = {G}_{1} \cup {G}_{2} \cup {G}_{3} \cup \ldots \cup {G}_{m}$ is given by,

barx=(Sigma_(i=1)^(i=m) n_ibarx_i)/(Sigma_(i=1)^(i=m) n_i.

We have, ${n}_{1} = 3 , {n}_{2} = 4 , {n}_{3} = 5 , {\overline{x}}_{1} = 7 , {\overline{x}}_{2} = 8 , {\overline{x}}_{3} = 11$.

Hence, $\overline{x} = \frac{3 \cdot 7 + 4 \cdot 8 + 5 \cdot 11}{3 + 4 + 5} = \frac{108}{12} = 9$, as derived by

Lyn. !

Enjoy Maths.!

Aug 11, 2016

$9$

#### Explanation:

To work out the mean (what we understand as the 'average') we do the following;

Mean = (" sum (total) of all the values")/("the number of values")" OR " " M=T/N

As with other formulae in this form, we have 3 ways in which it can be used.

$M = \frac{T}{N} \text{ "T = MxxN" } N = \frac{T}{M}$

Remember that we may add Totals , we may add Numbers , but we may NOT ADD MEANS!

The mean of 3 numbers is 7 .
$\text{There are 3 numbers, the mean is 7," rArr "the total is } 3 \times 7 = 21$

The mean of 4 numbers is 8
$\text{There are 4 numbers, the mean is 8," rArr "the total is } 4 \times 8 = 32$

The mean of 5 number is 11
$\text{There are 5 numbers, mean is 11" rArr "the total is } 5 \times 11 = 55$

We can add numbers: $3 + 4 + 5 = 12$

We can add totals: $21 + 32 + 55 = 108$

We can find the mean of the 12 numbers: $M = \frac{108}{12} = 9$