Find the mean and variance for the following frequency distribution.

Classes	0–30	30–60	60–90	90–120	120–150	150–180	180–210
Frequency (f)	2	3	5	10	3	5	2

We are given the cumulative frequency distribution. So, first we will prepare the frequency distribution as given below:

Class Interval	Cumulative frequency	Mid-value	Frequency	ui = xi - 67.5/15	fiui	fiui2
0 – 15	12	7.5	12	-4	-48	192
15 – 30	30	22.5	18	-3	-54	162
30 – 45	65	37.5	35	-2	-70	140
45 – 60	107	52.5	42	-1	-42	42
60 – 75	157	67.5	50	0	0	0
75 – 90	202	82.5	45	1	45	45
90 – 105	222	97.5	20	2	40	80
105 – 120	230	112.5	8	3	24	72
	230				-105	733

Σf_i = 230

Σf_iu_i = -105

Σf_iu_i² = 733

∴ Mean = a + h(¹/_NΣf_iu_i) = 67.5 + 15(^-105/₂₃₀) = 60.65

Variance (σ²) = h²[¹/_NΣf_iu_i² - (¹/_NΣf_iu_i)²]

= 225[⁷³³/₂₃₀ - (^-105/₂₃₀)²] = 671.51

Standard deviation = √Variance = √671.51 = 25.91