Question

Yoga is an ancient practice which is a form of meditation and exercise. By practising yoga, we not only make our body healthy but also achieve inner peace and calmness. The International Yoga Day is being celebrated on the 21st of June every year since 2015. To promote yoga, a society in Pune organised a 7-day yoga camp in their society. The number of people of different age groups who enrolled for this camp is given as follows: Age group 15–25 25–35 35–45 45–55 55–65 65–75 75–85 Number of people 8 10 15 25 40 24 18

Based on the above information, answer the following questions.

1. Which age group had the highest number of participants enrolled for the camp?
2. Find the modal age of people enrolled for the camp.
3. Find the median age of people enrolled for the camp.
4. If x more people of age group 55–65 had enrolled for the camp, the mean age would have been 57. Find the value of x.

Answer 1

i. Age group with the highest number of participants

The highest frequency is 40, which corresponds to the age group 55 – 65.

ii. Modal age of people enrolled

The modal class is 55 – 65 since it has the maximum frequency.

Lower limit of modal class, \[l = 55\]

Class width, \[h = 10\]

Frequency of modal class, \[f_1 = 40\]

Frequency of preceding class, \[f_0 = 25\]

Frequency of succeeding class, \[f_2 = 24\]

Formula: \[\text{Mode} = l + \left[\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right] \times h\]

Calculation: \[\text{Mode} = 55 + \left[\frac{40 - 25}{2(40) - 25 - 24}\right] \times 10\]

\[\text{Mode} = 55 + \left[\frac{15}{80 - 49}\right] \times 10\] 

\[\text{Mode} = 55 + \left(\frac{150}{31}\right)\] 

\[\text{Mode} = 55 + 4.84 = 59.84\text{ years}\]

iii. Median age of people enrolled

Cumulative frequency table:

Age group	Number of people \((f_i)\)	Cumulative frequency \((cf)\)
15 – 25	8	8
25 – 35	10	18
35 – 45	15	33
45 – 55	25	58
55 – 65	40	98
65 – 75	24	122
75 – 85	18	140
Total	\[N = \sum f_i = 140\]

\[\frac{N}{2} = \frac{140}{2} = 70\]

The cumulative frequency just greater than 70 is 98, which falls in the class 55 – 65.

Hence, the median class is 55 – 65:

\[l = 55\]

\[h = 10\]

\[f = 40\]

\[cf = 58\]

Formula: \[\text{Median} = l + \left[\frac{\frac{N}{2} - cf}{f}\right] \times h\]

Calculation: \[\text{Median} = 55 + \left[\frac{70 - 58}{40}\right] \times 10\]

\[\text{Median} = 55 + \left(\frac{12}{4}\right)\]

\[\text{Median} = 55 + 3 = 58\text{ years}\]

iv. Value of x

When x more participants join the age group 55 – 65, its frequency becomes 40 + x.

Age group	Class mark \((x_i)\)	Frequency \((f_i)\)	\(f_i x_i\)
15 – 25	20	8	160
25 – 35	30	10	300
35 – 45	40	15	600
45 – 55	50	25	1250
55 – 65	60	40 + x	2400 + 60x
65 – 75	70	24	1680
75 – 85	80	18	1440
Total		\[\sum f_i = 140 + x\]	\[\sum f_i x_i = 7830 + 60x\]

Given: \[\text{New Mean} = 57\]

Equation: \[\frac{\sum f_i x_i}{\sum f_i} = 57\] 

\[\frac{7830 + 60x}{140 + x} = 57\] 

\[7830 + 60x = 57(140 + x)\] 

\[7830 + 60x = 7980 + 57x\] 

\[60x - 57x = 7980 - 7830\] 

\[3x = 150\] 

\[x = 50\]