NCERT Example 2 — Statistics

NCERT Example

Example 2

mean · grouped data · direct assumed mean step deviation

The percentage distribution of female teachers in primary schools of rural areas of various States and Union Territories is given below. Find the mean percentage by the direct, assumed-mean and step-deviation methods.

Female teachers (%)	States/U.T. (fᵢ)
15–25	6
25–35	11
35–45	7
45–55	4
55–65	4
65–75	2
75–85	1

Solution

Board-exam working:

The class marks are the midpoints of the intervals. Take the assumed mean as 50 and class size as 10.

Class marks and deviations (a = 50, h = 10)
Class	fᵢ	xᵢ	dᵢ	uᵢ
15–25	6	20	−30	−3
25–35	11	30	−20	−2
35–45	7	40	−10	−1
45–55	4	50	0	0
55–65	4	60	10	1
65–75	2	70	20	2
75–85	1	80	30	3

Products used by the three methods
Class	fᵢxᵢ	fᵢdᵢ	fᵢuᵢ
15–25	120	−180	−18
25–35	330	−220	−22
35–45	280	−70	−7
45–55	200	0	0
55–65	240	40	4
65–75	140	40	4
75–85	80	30	3

\sum f_i=35,\quad \sum f_ix_i=1390,\quad \sum f_id_i=-360,\quad \sum f_iu_i=-36

Direct method:

\bar{x}=\frac{1390}{35}=39.714\ldots

Assumed-mean method:

\bar{x}=50+\frac{-360}{35}=39.714\ldots

Step-deviation method:

\bar{x}=50+10\left(\frac{-36}{35}\right)=39.714\ldots

Final answer

\boxed{\bar{x}\approx39.71\%}