Calculate the mean and variance for the following data:

 

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 (fᵢ)	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	57	37.5	27	−2	−54	108
45 – 60	107	52.5	50	−1	−50	50
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
Total			Σfᵢ = 230		Σfᵢuᵢ = −105	Σfᵢuᵢ² = 733

Given: a = 67.5, h = 15, N = 230, Σfᵢuᵢ = −105, Σfᵢuᵢ² = 733

Mean:
Mean = a + h(Σfᵢuᵢ / N)
= 67.5 + 15(−105 / 230)
= 67.5 − 6.85 = 60.65

Variance:
σ² = h²[(Σfᵢuᵢ² / N) − (Σfᵢuᵢ / N)²]
= 225[(733 / 230) − (−105 / 230)²]
= 225[3.1870 − 0.2025] = 671.51

Standard Deviation:
√Variance = √671.51 = 25.91