Question

Find the mean and variance of frequency distribution given below:

xi:	1 ≤ x < 3	3 ≤ x < 5	5 ≤ x < 7	7 ≤ x < 10
fi:	6	4	5	1

Answer 1

xi	Mid-Values(yi)	yi2	fi	fi yi	fi yi2
1–3	2	4	6	12	24
3–5	4	16	4	16	64
5–7	6	36	5	30	180
7–10	8.5	72.25	1	8.5	72.25
			`sum f_i = 16`	\[\sum_{} f_i y_i = 66 . 5\]	\[\sum_{}$ f_i {y_i}^2 = 340 . 25\]

Therefore,

Mean =

\[\frac{\sum_{} f_i y_i}{\sum_{} f_i} = \frac{66 . 5}{16} = 4 . 16\]

Variance =\[\left( \frac{1}{N} \sum_{} f_i y_i^2 \right) - \left( \frac{1}{N} \sum_{} f {}_i y {}_i \right)^2 = \frac{1}{16} \times 340 . 25 - \left( \frac{1}{16} \times 66 . 5 \right)^2 = 21 . 26 - 17 . 22 = 4 . 04\]