Question

The arithmetic mean of 12 observations is 7.5. If the arithmetic mean of 7 of these observations is 6.5, then find the mean of the remaining observations.

Answer 1

Hint: The formula for arithmetic mean for a given set of $n$ observations $\{ {a_1},{a_2}, \ldots ,{a_n}\}$ is
\[A.M. = \dfrac{1}{n}\sum\limits_{i = 1}^n {{a_i}}\]
We will use this formula to obtain the sum of all observations. Then, we will use this formula again to get the sum of the 7 observations. So, now we can get the sum of the remaining 5 observations. After obtaining the sum of the remaining 5 observations, we will use this same formula again to find the mean of these remaining observations.

Complete step-by-step answer:
The arithmetic mean formula is
\[A.M. = \dfrac{1}{n}\sum\limits_{i = 1}^n {{a_i}}\]
We have a total of 12 observations. The mean of these 12 observations is given as 7.5. Now we will first use the arithmetic formula for 12 observations. We will substitute the values for $A.M.$ and $n$ as follows,
\[7.5 = \dfrac{1}{{12}}\sum\limits_{i = 1}^{12} {{a_i}}\]
Let the sum of all 12 observations be $p$. So we have,
\[7.5=\dfrac{1}{12}\times p\]
Therefore, we get the value of $p$ by solving the above equation as follows,
\[\begin{align}
  & p=7.5\times 12 \\
 & =90
\end{align}\]
Now, the arithmetic mean of 7 of these observations is 6.5. Let the sum of these 7 observations be $q$. We will substitute these values in the arithmetic mean formula. We will get the following equation,
\[6.5 = \dfrac{1}{7}q\]
So, the sum of the 7 observations is calculated by solving the above equation as follows,
\[\begin{align}
  & q=6.5\times 7 \\
 & =45.5
\end{align}\]
Now, the remaining observations are $12 - 7 = 5$. The sum of the remaining 5 observations is $p - q$. To find the mean of the remaining 5 observations, we will substitute these values in the arithmetic mean formula. We will obtain the following equation,
\[A.M.=\dfrac{1}{5}(p-q)\]
We know the values of $p$ and $q$. Substituting these values in the above equation, we get
\[A.M.=\dfrac{1}{5}\times (90-45.5)\]
Solving this equation, we will get the mean of the remaining 5 observations as follows,
\[\begin{align}
  & A.M.=\dfrac{1}{5}\times 44.5 \\
 & =8.9
\end{align}\]
So, the arithmetic mean of the remaining 5 observations is 8.9.

Note: While using the same formula more than once, it is easy to mix up the values obtained. Hence, one should avoid this confusion by using different variables for every iteration. The important part in this question is understanding the meaning of the arithmetic mean and its formula.