Question

Nelly computed the mean of $5$ numbers to be $7$. When she checks her work, she realizes she misread $3$ as $8$. The actual mean of the data set is .

Answer 1

Nelly misread the actual number $3$ as $8$, and got the mean of $5$ numbers as $7$.
Consider, the other four numbers are [ ], [ ], [ ] and [ ].

As the mean of the data set $\left\{\right[],[],[],[],8\}$ is $7$, we can write

$\frac{\left[\right]+\left[\right]+\left[\right]+\left[\right]+8}{5}=7$

$\Rightarrow \frac{\left[\right]+\left[\right]+\left[\right]+\left[\right]+8}{5}\times 5=7\times 5$
(multiplying $5$ in both sides)

$\Rightarrow \left[\right]+\left[\right]+\left[\right]+\left[\right]+8=35$

​​​​​​​$\Rightarrow \left[\right]+\left[\right]+\left[\right]+\left[\right]+8-8=35-8$
(substracting $8$ from both sides)

​​​​​​​$\Rightarrow \left[\right]+\left[\right]+\left[\right]+\left[\right]=27$

$\text{Need to find:}$ Actual mean of the data set $\left\{\right[],[],[],[],3\}$, i.e.,

$\frac{\left[\right]+\left[\right]+\left[\right]+\left[\right]+3}{5}$

$=\frac{27+3}{5}(\because []+[]+[]+[]=27)$

$=\frac{30}{5}=6$

$\therefore$ The actual mean of the data set is $6$.