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.

detailed solution

1

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