Question

Arithmetic mean of a certain number of observations was 60.8. Five new observations 85, 

60, 73, 81, 90 were added to the group. Now the arithmetic mean of all the observations 

becomes 65.05. Find how many observations were there originally.

Answer 1

Before adding the new observations to the new group, the mean was 60.8 and it is derived from using the formula given as,

$\bar{x}=Y/n$, where $Y=\displaystyle\sum^n_{i=1}(x_i)$ and $n$ is the sample size.

Now,

$\bar{x}_1=Y/n=60.8.........(i)$

After adding the new observations the mean becomes 65.05 and the sample size is $n+5$and the numerator for the mean becomes $\displaystyle\sum^n_{i=1}(x_i)+(85+60+73+81+90)=\displaystyle\sum^n_{i=1}(x_i)+389$ . We can write this as,

$\bar{x}_2=(\displaystyle\sum^n_{i=1}(x_i)+389)/(n+5)=(Y+389)/(n+5)=65.05.......(ii)$

From equation (i), let us make $n$ the subject of the formula, therefore,

$n=Y/\bar{x}=Y/60.8$. Substituting for the value of $n$ in equation (ii),

$(Y+389)/((Y/60.8)+5)=65.05\equiv (Y+389)=65.05(Y/60.8)+5)$

This can also be written as,

$Y+389=1.06990132Y+325.05........(iii)$

Collecting like terms in equation (iii),

$1.06990132Y-Y=389-325.05$

$0.06990132Y=63.75\implies Y=911.999945\approx 912$

We can determine the original sample size by putting the value of $Y$ found above into equation (i).

$Y/n=60.8$

Now,

$912/n=60.8\implies n=14.9999991\approx 15$

Therefore, the number of observations originally were 15.