The following is the cumulative frequency distribution (of less than type) of $1000$ persons each of age $20$ $years$ and above. Determine the mean age.

Age (years)	$30$	$40$	$50$	$60$	$70$	$80$
Cumulative Frequency	$100$	$220$	$350$	$750$	$950$	$1000$

(N/A) First,we convert the cumulative frequency distribution into a standard frequency distribution. The class interval starts from $20-30$ as the age is $20$ and above.

Class Interval	Frequency $(f_i)$	Class mark $(x_i)$	$u_i = \frac{x_i - 45}{10}$	$f_i u_i$
$20-30$	$100$	$25$	$-2$	$-200$
$30-40$	$120$	$35$	$-1$	$-120$
$40-50$	$130$	$45$	$0$	$0$
$50-60$	$400$	$55$	$1$	$400$
$60-70$	$200$	$65$	$2$	$400$
$70-80$	$50$	$75$	$3$	$150$
Total	$\sum f_i = 1000$	-	-	$\sum f_i u_i = 630$

Using the step-deviation method: $\text{Mean} (\bar{x}) = a + h \left( \frac{\sum f_i u_i}{\sum f_i} \right)$
Here,$a = 45$,$h = 10$,$\sum f_i u_i = 630$,and $\sum f_i = 1000$.
$\bar{x} = 45 + 10 \left( \frac{630}{1000} \right) = 45 + 6.3 = 51.3$.
Thus,the mean age is $51.3$ years.