Question

On average, 80 patients get admitted into a nursing home in a day. The ages of the patients admitted and their number are as given below:

Age (in years)	10-20	20-30	30-40	40-50	50-60
No. of patients	13	23	25	14	5

Find:

1. The average age for which maximum cases occur.
2. The upper limit of modal class.
3. The mean of the given data.

Answer 1

i. The average age for which maximum cases occur is the mode of the distribution.

The maximum frequency is 25, so the modal class is 30 - 40.

Here, l = 30, f₁ = 25, f₀ = 23, f₂ = 14 and h = 10.

By formula,

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

\[{} = 30 + \frac{25 - 23}{2(25) - 23 - 14} \times 10\]

\[{} = 30 + \frac{2}{50 - 37} \times 10\]

\[{} = 30 + \frac{2}{13} \times 10\]

\[{} = 30 + \frac{20}{13}\]

= 30 + 1.54

= 31.54

Therefore, the average age for which maximum cases occur is 31.54 years (approx.).

ii. The modal class is 30 - 40.

Therefore, the upper limit of the modal class is 40.

iii. Mean of the data:

Age (in years)	Class-mark (x)	No. of patients (f)	fx
10 - 20	15	13	195
20 - 30	25	23	575
30 - 40	35	25	875
40 - 50	45	14	630
50 - 60	55	5	275
Total		Σf = 80	Σfx = 2550

By formula,

\[\text{Mean} = \frac{\Sigma fx}{\Sigma f} = \frac{2550}{80} = 31.875 \approx 31.88.\]

Therefore, the mean of the given data is 31.88 years (approx.).