Question

How do you find the mean : $y-2a,y-a,y,y+a,y+2a$ ?

Answer 1

Hint: A mean is the simple mathematical average of a set of two or more numbers or an expression consisting of variables and constants . The basic way to find out the mean is to divide the sum of terms with the total number of terms. Instead of using the word terms, we should use the word observations when we are in the topic of statistics. Observations are nothing but another word for terms. In this case, there are $5$ terms.

Complete step-by-step answer:
Mean is usually denoted with M.
Let us see the formula of mean.
Mean (M) = $\dfrac{\text{sum of observations}}{\text{number of observations}}$
Here, our observations are $y-2a,y-a,y,y+a,y+2a$.
Let us add them.
$\Rightarrow y-2a+y-a+y+y+a+y+2a$
Let us group all the $y$terms together.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow y-2a+y-a+y+y+a+y+2a \\
 & \Rightarrow y+y+y+y+y-2a-a+a+2a \\
\end{align}$
Now, let us add all the $y$terms.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow y-2a+y-a+y+y+a+y+2a \\
 & \Rightarrow y+y+y+y+y-2a-a+a+2a \\
 & \Rightarrow 5y-2a-a+a+2a \\
\end{align}$
We can clearly see that the other observations or terms are being added and subtracted with the same magnitude. So they sum up to be $0$.
Upon solving further , we get the following :
$\begin{align}
  & \Rightarrow y-2a+y-a+y+y+a+y+2a \\
 & \Rightarrow y+y+y+y+y-2a-a+a+2a \\
 & \Rightarrow 5y-2a-a+a+2a \\
 & \Rightarrow 5y \\
\end{align}$
Upon solving , we get the sum of observations to be $5y$ .
We have $5$ observations.
Now, let us substitute the results that we got in the formula of mean and calculate for it.
Upon substituting, we get the following :
$\Rightarrow $ Mean (M) = $\dfrac{\text{sum of observations}}{\text{number of observations}}$
$\Rightarrow $ Mean (M) = $\dfrac{5y}{5}=y$ .
$\therefore $ Hence, the mean of the $5$ observations which are $y-2a,y-a,y,y+a,y+2a$ is $y$.

Note: It is very important to know the formulae of mean, median and mode. The questions involving statistics are very time-taking. So, a lot of practice must be put in so as to complete the question in exam on time. The basic intuition behind mean is used for comparison of quantities. It is to know where one quantity stands among a lot of other quantities.