Question

The mean of four observations is $12$. If first, second, third and fourth observations are decreased by $1,2,3$ and $4$ respectively, then find the new mean?

Answer 1

From the question, we have to find the mean of the given data. First, we have to frame a mathematical expression for the given data. Then, we have to find the new mean by data handling.
Data handling is an art. Sometimes the raw data (data as they are) will not be useful to get the required information. In order to get proper useful information, we have to process very important useful information from the given data.
Formula Used:
The mean is the meaning of an average. The process of sharing out equally is the basis of average. We call the averaged or quantity as the arithmetic average (or arithmetic mean or simply average or mean).
The mean of several items is the value equally shared out among the items.
${\text{Mean = }}\dfrac{{{\text{Total of all items}}}}{{{\text{Number of items}}}}$.
$\Rightarrow {\text{Total of all items = Mean}} \times {\text{Number of items}}$
Let us consider the four observations to be ${\text{a,b,c}}$ and ${\text{d}}$.
Let the total of four observations be ${\text{a}} + {\text{b}} + {\text{c}} + {\text{d}}$.
Let the first observation can be decreased by one as ${\text{a}} - 1$.
Let the second observation can be decreased by two as ${\text{b}} - 2$.
Let the third observation can be decreased by three as ${\text{c}} - 3$.
Let the fourth observation can be decreased by four as ${\text{d}} - 4$.
Let the total of above new four observations be expressed as $\left( {{\text{a}} - 1} \right) + \left( {{\text{b}} - 2} \right) + \left( {{\text{c}} - 3} \right) + \left( {{\text{d}} - 4} \right)$.

Complete step by step answer:
-According to the question, we have the mean of four observations. Now, we are going to write in the mathematical expression.
${\text{Mean = }}\dfrac{{{\text{Total of four observations}}}}{{{\text{Number of observations}}}}$
Here, the number of observation is $4$ and the given mean of four observations are $12$
Substitute the above given data in the formula. Then, we get
$12 = \dfrac{{{\text{a}} + {\text{b}} + {\text{c}} + {\text{d}}}}{4}$
$\Rightarrow {\text{a}} + {\text{b}} + {\text{c}} + {\text{d}} = 12 \times 4$
$\Rightarrow {\text{a}} + {\text{b}} + {\text{c}} + {\text{d}} = 48$
Now, we have to find the new mean from the given data.
Let the total of above new four observations be expressed as $\left( {{\text{a}} - 1} \right) + \left( {{\text{b}} - 2} \right) + \left( {{\text{c}} - 3} \right) + \left( {{\text{d}} - 4} \right)$ and this term can also be written as
$\Rightarrow {\text{a}} + {\text{b}} + {\text{c}} + {\text{d}} + \left( { - 1 - 2 - 3 - 4} \right)$
$\Rightarrow {\text{a}} + {\text{b}} + {\text{c}} + {\text{d}} + \left( { - 10} \right)$
$\Rightarrow {\text{a}} + {\text{b}} + {\text{c}} + {\text{d}} - 10$
Already we know that the total of the four observations are $48$. That is ${\text{a}} + {\text{b}} + {\text{c}} + {\text{d}} = 48$ and now, we are going to substitute them in the above total of new observation. Then, we get
$\Rightarrow 48 - 10 = 38$
Therefore, the total of the new four observations are $38$.
The number of observations is still the same $4$ here. Now, we have to find the new mean for the above four new observations.
${\text{Mean}}\left( {{\text{for new four observations}}} \right){\text{ = }}\dfrac{{{\text{Total of new four observations}}}}{{{\text{Number of observations}}}}$
Mean $= \dfrac{{38}}{4}$
Mean $= 9.5$
$\therefore$ The new mean $9.5$.

Note:
Mean is an essential concept in mathematics and statistics. The mean is the average or the most common value in a collection of numbers. In statistics, it is a measure of central tendency of a probability distribution along median and mode. It is also referred to as an expected value.
The given problem is an easy one to solve. We have to be careful when expressing the given data in the mathematical form. Then, we should concentrate on further simple calculations on finding the new mean by using the mean formula.