In a hospital, the ages of diabetic patients were recorded as follows. Find the median age. [CBSE 2014]

Age (in years) 0−15 15−30 30−45 45−60 60−75

Number of patients 5 20 40 50 25

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Solution

We prepare the cumulative frequency table, as shown below:

Age (in years) Number of patients (f_i) Cumulative Frequency (cf)

0−15 5 5

15−30 20 25

30−45 40 65

45−60 50 115

60−75 25 140

Total N = ∑ f_i = 140

Now, N = 140 $\Rightarrow \frac{N}{2} = 70$ .

The cumulative frequency just greater than 70 is 115 and the corresponding class is 45−60.

Thus, the median class is 45−60.

∴ l = 45, h = 15, f = 50, N = 140 and cf = 65.

Now,

$Median = l + (\frac{\frac{N}{2} - c f}{f}) \times h = 45 + (\frac{\frac{140}{2} - 65}{50}) \times 15 = 45 + (\frac{70 - 65}{50}) \times 15 = 45 + 1.5 = 46.5$

Hence, the median age is 46.5 years.

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Form a 'less than type' cumulative frequency distribution. [CBSE 2013]

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Age (in years)	0−15	15−30	30−45	45−60	60−75
Number of patients	5	20	40	50	25