Question

A group of $10$ items has an arithmetic mean of $6$, if the arithmetic mean of $4$ of those items is $7.5$. Then what is the meaning of the remaining elements?
${\text{(A) 6}}{\text{.5}}$
${\text{(B) 5}}{\text{.5}}$
${\text{(C) 4}}{\text{.5}}$
${\text{(D) 5}}{\text{.0}}$

Answer 1

Hint: Here we have to find the mean of the remaining elements. By using the data, we can find the sums of the items are $10$ and $4$ by using the formula. Then we find the total number of items and sum of the item. Finally we use the formula and find the required answer.

Formula used: $Mean = \dfrac{{{\text{Sum of terms}}}}{{{\text{number of terms}}}}$

Complete step-by-step solution:
It is given that the question stated as the mean of all the terms is $6$ and since there are total $10$ terms in the distribution, using the formula of mean we can write it as:
$ \Rightarrow 6 = \dfrac{{{\text{Sum of terms}}}}{{10}}$
On cross multiplying the equation, we get:
$ \Rightarrow 6 \times 10 = {\text{Sum of terms}}$
On simplifying we get:
$ \Rightarrow 60 = {\text{Sum of terms}}$
Therefore, we know the sum of all the $10$ terms in the distribution is $60$
Also, the mean of the any $4$ numbers from the distribution is $7.5$ therefore, using the formula of mean we can write it as:
$ \Rightarrow 7.5 = \dfrac{{{\text{Sum of terms}}}}{4}$
On cross multiplying the above equation, we get:
$ \Rightarrow 7.5 \times 4 = {\text{Sum of terms}}$
On simplifying we get:
$ \Rightarrow 30 = {\text{Sum of terms}}$
Therefore, we know the sum of all the $4$ terms in the distribution is $30$.
Now since there are total $10$ terms in the distribution which have a sum of $60$ out of which we know the sum of terms of $4$ terms is $30$ therefore the sum of the remaining terms is:
$ = 60 - 30$
On simplifying we get:
Sum of the terms$ = 30$
Now since we subtracted the sum of $4$ terms from the total number of terms, the remaining numbers of terms are:
$ = 10 - 4$
On simplifying we get:
Number of terms$ = 6$
Therefore, we have to use the formula of mean, the mean of the remaining terms in the distribution is:
$Mean = \dfrac{{30}}{6}$
On dividing the terms we get:
$Mean = 5$, which is the required answer.

Therefore, the correct option is $(D)$.

Note: Mean is the most commonly used measure of central tendency and it is usually considered to be the measure of it. Mean should not be used when the data is non-numeric or when there are extreme values in the distribution.
Mode is to be used when the given distribution is nominal, which implies that the class or category is not based on numbers for example: city, age, gender etc.