Question

The following data was obtained from a survey. Find the mean of this data

Age (in years)	Number of persons
Less than 10	5
Less than 20	12
Less than 30	16
Less than 40	25
Less than 50	45
Less than 60	52
Less than 70	60
Less than 80	63

Answer 1

Hint: We will first create a table of class interval and frequency where class interval of age is taken as $0-10,10-20,20-30......70-80$.
Now we know that now once we have class interval we will find a class mark.
Classmark of interval a to b is given by $\dfrac{a+b}{2}$ .
Now the mean is given by $\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}$ where ${{f}_{i}}$ are frequencies and ${{X}_{i}}$ is class marks.

Now first let us consider the table.

Age (in years)	Number of persons
Less than 10 (0 – 10)	5
Less than 20 (0 – 20)	12
Less than 30 (0 – 30)	16
Less than 40 (0 – 40)	25
Less than 50 (0 – 50)	45
Less than 60 (0 – 60)	52
Less than 70 (0 – 70)	60
Less than 80 (0 – 80)	63

Now we first want to the data according to class Intervals $0-10,10-20,20-30......70-80$ .
Now we know that if there are 12 people with people age less than 20 and 5 people with age less than 10. Then the number of people between age 10 and 20 are 12 – 5 = 7
Similarly, we can find the data with class interval of $0-10,10-20,20-30......70-80$ .

Age (in years) (CLASS INTERVAL)	Number of persons (FREQUENCY)
(0 – 10)	5
(10 – 20)	12 – 5 = 7
(20 – 30)	16 – 12 = 4
(30 – 40)	25 – 16 = 9
(40 – 50)	45 – 25 = 20
(50 – 60)	52 – 45 = 7
(60 – 70)	60 – 52 = 8
(70 – 80)	63 – 60 = 3

Now we know that class mark of interval a to b is given by $\dfrac{a+b}{2}$ .
Hence Now

Age (in years) (CLASS INTERVAL)	CLASS MARKS$\left( {{X}_{i}} \right)$	Number of persons (FREQUENCY) ${{f}_{i}}$	${{X}_{i}}{{f}_{i}}$
(0 – 10)	$\dfrac{0+10}{2}=5$	5	25
(10 – 20)	$\dfrac{10+20}{2}=15$	12 – 5 = 7	105
(20 – 30)	$\dfrac{20+30}{2}=25$	16 – 12 = 4	100
(30 – 40)	$\dfrac{30+40}{2}=35$	25 – 16 = 9	315
(40 – 50)	$\dfrac{40+50}{2}=45$	45 – 25 = 20	900
(50 – 60)	$\dfrac{50+60}{2}=55$	52 – 45 = 7	385
(60 – 70)	$\dfrac{60+70}{2}=65$	60 – 52 = 8	520
(70 – 80)	$\dfrac{70+80}{2}=75$	63 – 60 = 3	225
		$\sum{{{f}_{i}}=63}$	$\sum{{{f}_{i}}{{X}_{i}}=2575}$

Now mean of data is given by $\dfrac{\sum{{{f}_{i}}{{X}_{i}}}}{\sum{{{f}_{i}}}}=\dfrac{2575}{63}=40.87$
Hence the mean of given data is 40.87.

Note:
Now note that the given data is in cumulative frequency format hence we cannot directly calculate the mean hence first we find the frequency of each interval and then calculate mean