下面给出了一个柜台在 40 天内每天服务的顾客人数 $c$。

1. Find class midpoints and compute the weighted total

Class midpoints: $2.5,\ 7.5,\ 12.5,\ 17.5$

$\sum fx=11(2.5)+18(7.5)+9(12.5)+2(17.5)=310$

2. Divide by the total frequency to find the mean

Total frequency $=40$.

$\text{Mean}=\frac{310}{40}=7.75$

1. Use the class midpoints to find $\sum fx^2$

The class midpoints are $2.5,\ 7.5,\ 12.5,\ 17.5$.

$\sum fx^2=11(2.5)^2+18(7.5)^2+9(12.5)^2+2(17.5)^2$

$=68.75+1012.5+1406.25+612.5=3100$

2. Substitute into the standard deviation formula

Using $\sum f=40$ and the grouped-data mean $\bar{x}=7.75$,

$\sigma=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}^2}$

$=\sqrt{\frac{3100}{40}-(7.75)^2}$

$=\sqrt{17.4375}=4.1758\ldots$

$\sigma\approx 4.18$

1. Determine the new mean after adding 3 customers each day

Adding $3$ customers to each day's number increases every data value by $3$.

So the mean also increases by $3$.

$8.75+3=11.75$

2. Determine the new standard deviation after adding 3 customers each day

Adding the same number, $3$, to every data value does not change the spread of the data.

So the standard deviation remains unchanged.

$4.18$