The following table shows the ages of the patients admitted to a hospital during a year:

Age (in years)	5-15	15-25	25-35	35-45	45-55	55-65
Number of patients	6	11	21	23	14	5

Find the mode and the mean of the data given above. Compare and interpret the two
measures of central tendency.

For the modal class, let us the consider the class interval with highest frequency

Here, the greatest frequency = 23, so the modal class = 35 – 45,

l = 35,

class width (h) = 10,

f_m = 23,

f₁ = 21 and f₂ = 14

The formula to find the mode is

Mode = l+ [$\frac{(f_m-f_1)}{2f_m-f_1-f_2}$]×h

Substitute the values in the formula, we get

Mode = 35+[$\frac{(23-21)}{46-21-14}$]×10

Mode = 35+($\frac{20}{11}$) = 35+1.8

Mode = 36.8 year

So the mode of the given data = 36.8 year

Calculation of Mean:

First find the midpoint using the formula, x_i = $\frac{upper limit+lower limit}{2}$

The mean formula is

Mean = x̄ = $\frac{{\displaystyle \sum _{ }}{f}_i{x}_i}{\sum _{ }{f}_i}$

= $\frac{2830}{80}$

= 35.37 years

Therefore, the mean of the given data = 35.37 years