Question

The diameters of circles (in mm) drawn in a design are given below

Diameters	No. of children
33-36	15
37-40	17
41-44	21
45-48	22
49-52	25

Answer 1

Class Interval	Frequency \(f_i\)	\(mid-point\,x_i\)	\(y_i=\frac{x_i-42.5}{4}\)	\(f_i^2\)	\(f_iy_i\)	\(f_iy_1^2\)
32.5-36.5	15	34.5	-2	4	-30	60
36.5-40.5	17	38.5	-1	1	-17	17
40.5-44.5	21	42.5	0	0	0	0
44.5-48.5	22	46.5	1	1	22	22
48.5-52.5	25	50.5	2	4	50	100
	100				25	199

Here, N = 100, h = 4 

Let the assumed mean, A, be 42.5

Mean, \(\bar{x}=A\frac{\sum_{i=1}^5f_ix_i}{n}×h=42.5+\frac{25}{100}×4=43.5\) 

Variance (σ²) = \(\frac{h^2}{N^2}[N\sum_{i=1}^5f_iy_i^2-(\sum_{i=1}^5f_iy_i)^2]\)

\(=\frac{16}{10000}[100×199-(25)^2]\)

\(=\frac{16}{10000}[19900-625]\)

\(=\frac{16}{1000}×19275\)

\(=30.84\)

\(Standard\,\,deviation\.(σ) =5.55\)