Calculate the mean, median and mode of the following distribution:

Number	Frequency
5	1
10	2
15	5
20	6
25	3
30	2
35	1

The variates are already in ascending order. We construct the cumulative frequency table as under:

Total number of observations = 20, which is even.

By formula,

$\Rightarrow \text{Mean} = \dfrac{\sum\text{fx}}{\sum\text{f}} \\[1em] \Rightarrow \text{Mean} = \dfrac{390}{20} \\[1em] \Rightarrow \text{Mean} = 19.5$

By formula,

Median = $\dfrac{\dfrac{\text{n}}{2} \text{ th observation} + \Big(\dfrac{\text{n}}{2} + 1\Big) \text{ th observation}}{2}$

$= \dfrac{\dfrac{20}{2} \text{ th observation} + \Big(\dfrac{20}{2} + 1\Big) \text{ th observation}}{2} \\[1em] = \dfrac{10 \text{ th observation} + \Big(10 + 1\Big) \text{th observation}}{2} \\[1em] = \dfrac{10 \text{ th observation} + 11 \text{ th observation}}{2} \\[1em]$

All observations from 9th to 14th are equal, each = 20

Then,

Median = $\dfrac{20 + 20}{2} = \dfrac{40}{2}$ = 20.

As the variate 20 has maximum frequency 6, so mode = 20.

Hence, mean = 19.5, median = 20, mode = 20.