Question 1
The following table shows the ages of the patients admitted to a hospital during a year:
Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.
| Age (in years) | 5-15 | 15-25 | 25-35 | 35- 45 | 45-55 | 55-65 |
| Number of patients | 6 | 11 | 21 | 23 | 14 | 5 |
Solution:
To find out the modal class, let us the consider the class interval with high frequency.
Here, the greatest frequency = 23, so the modal class = 35 – 45,
Lower limit of modal class = l = 35,
class width (h) = 10,
$f_m$ = 23,
$f_{1}$ = 21 and $f_{2}$ = 14
The formula to find the mode is
| Mode = $l + [ \dfrac{(f_m – f_{1})}{(2f_m – f_{1} – f_{2})}] × h$ |
Substitute the values in the formula, we get
Mode = 35+$[\dfrac{(23-21)}{(46-21-14)}]$×10
= 35 + $\dfrac{20}{11}$
= 35 + 1.8
= 36.8 years
So the mode of the given data = 36.8 years
Calculation of Mean:
First find the midpoint using the formula, $x_{i} = \dfrac{(upper \: limit + lower \: limit)}{2}$
| Class Interval | Frequency $(f_{i})$ | Mid-point $(x_{i})$ | $f_{i}x_{i}$ |
|---|---|---|---|
| 5-15 | 6 | 10 | 60 |
| 15-25 | 11 | 20 | 220 |
| 25-35 | 21 | 30 | 630 |
| 35-45 | 23 | 40 | 920 |
| 45-55 | 14 | 50 | 700 |
| 55-65 | 5 | 60 | 300 |
| Sum $f_i$ = 80 | Sum $f_{i}x_{i}$ = 2830 |
The mean formula is
| Mean =$ \overline{x} = \dfrac{∑f_{i}x_{i} }{∑f_{i}}$ |
= $\dfrac{2830}{80}$
= 35.375 years
Therefore, the mean of the given data = 35.375 years