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

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

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

We will find mean by direct method.

By formula,

Class mark = $\dfrac{\text{Upper limit + Lower limit}}{2}$

By formula,

Mean = $\dfrac{Σf$ixi}{Σf_i} = \dfrac{2830}{80} = 35.375

By formula,

Mode = l + $\Big(\dfrac{f$1 - f0}{2f1 - f0 - f_2}\Big) \times h

Here,

1. Class size is h.

2. The lower limit of modal class is l

3. The Frequency of modal class is f₁.

4. Frequency of class preceding modal class is f₀.

5. Frequency of class succeeding the modal class is f₂.

Class 35 - 45 has the highest frequency.

∴ It is the modal class.

∴ l = 35, f₁ = 23, f₀ = 21, f₂ = 14 and h = 10.

Substituting values we get :

$\text{Mode} = 35 + \Big(\dfrac{23 - 21}{2 \times 23 - 21 - 14}\Big) \times 10 \\[1em] = 35 + \dfrac{2}{46 - 35} \times 10 \\[1em] = 35 + \dfrac{20}{11} \\[1em] = 35 + 1.8 \\[1em] = 36.8$

Since, mode = 36.8

∴ Maximum number of patients admitted in the hospital are of the age 36.8 years.

Since, mean = 35.37

∴ Average the age of a patient admitted to the hospital is 35.37 years.