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

Find :

(i) The average age for which maximum cases occur.

(ii) The upper limit of modal class.

(iii) The mean of the given data.

(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 + \dfrac{f$1 - f0}{2f1 - f0 - f_2} \times h \\[1em] = 30 + \dfrac{25 - 23}{2(25) - 23 - 14} \times 10 \\[1em] = 30 + \dfrac{2}{50 - 37} \times 10 \\[1em] = 30 + \dfrac{2}{13} \times 10 \\[1em] = 30 + \dfrac{20}{13} = 30 + 1.54 = 31.54.

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

(ii) The modal class is 30 - 40.

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

(iii) Mean of the data :

By formula,

Mean = $\dfrac{Σfx}{Σf} = \dfrac{2550}{80}$ = 31.875 ≈ 31.88.

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