Question

The monthly income of a group of 320 employees in a company is given below:

Monthly income (in ₹)	No. of Employees
6000 - 7000	20
7000 - 8000	45
8000 - 9000	65
9000 - 10000	95
10000 - 11000	60
11000 - 12000	30
12000 - 13000	5

Draw an ogive the given distribution on a graph sheet taking 2 cm = Rs. 1000 on one axis and 2 cm = 50 employees on the other axis. From the graph determine:

1. the median wage
2. the number of employees whose income is below Rs. 8500.
3. if the salary of a senior employee is above Rs. 11500, find the number of senior employees in the company.
4. the upper quartile.

Answer 1

Monthly Income	No. of Employees	c.f.
6000 - 7000	20	20
7000 - 8000	45	65
8000 - 9000	65	130
9000 - 10000	95	225
10000 - 11000	60	285
11000 - 12000	30	315
12000 - 13000	5	320

Here, n = 320

1. Media  =  `(n/2)^(th)` term = 160^th term

From the graph, the corresponding x coordinate is 9400

Median wage = 9400 approx

2. The number of employees whose income is below Rs. 8500 = 95 (approx.)

3. The number of senior employees whose salary is above Rs. 11500

= 320 – 305

= 15 (approx.)

4. The upper quartile `Q_3 = ((3n)/4)^"th"` term = 240^th term

From the graph, the corresponding x co-ordinate is 10,300 (approx.)

Answer 2

Monthly	No. of Employees	c.f.
6000 - 7000	20	20
7000 - 8000	45	65
8000 - 9000	65	130
9000 - 10000	95	225
10000 - 11000	60	285
11000 - 12000	30	315
12000 - 13000	5	320

i. From the graph, the median wage = 160.5 (appro.)

ii. The number of employees whose income is below ₹ 8500 = 90 (appro.)

iii. The number of senior employees whose salary is above ₹ 11500 = 20 (appro.)

iv. The upper quartile Q₃ = 240 (appro.)

Answer 3

To draw the Less Than Ogive, we use the upper class limits and cumulative frequencies:

Monthly Income (₹)	No. of Employees (f)	Cumulative Frequency (cf)	Points to Plot
6000 - 7000	20	20	(7000, 20)
7000 - 8000	45	65	(8000, 65)
8000 - 9000	65	130	(9000, 130)
9000 - 10000	95	225	(10000, 225)
10000 - 11000	60	285	(11000, 285)
11000 - 12000	30	315	(12000, 315)
12000 - 13000	5	320	(13000, 320)

1. Calculate cumulative frequencies

First, we find the cumulative totals for each bracket to plot the curve.

Total employees (N) = 320.

The points are plotted using the Upper Limit on the X-axis and Cumulative Frequency on the Y-axis.

2. Determine median wage

The median is the `(N/2)^(th)` term, which is the 160^th employee.

Locate 160 on the Y-axis.

Draw a horizontal line to the curve, then drop a perpendicular to the X-axis.

Result: The median wage is approximately ₹ 9,300 to ₹ 9,460.

3. Employees income below ₹ 8,500

Locate ₹ 8,500 on the X-axis.

Draw a vertical line up to the curve, then a horizontal line to the Y-axis.

Result: There are approximately 95 to 98 employees with an income below ₹ 8,500.

4. Number of senior employees

Senior employees earn above ₹ 11,500.

Find the cumulative frequency for ₹ 11,500 on the graph. This is approximately 300 to 305 employees those earning below ₹ 11,500.

Subtract this from the total: 320 – 300 = 20.

Result: There are approximately 15 to 20 senior employees.

5. Find the upper quartile (Q₃)

The upper quartile is the `((3N)/4)^(th)` term, which is the 240^th employee.

Locate 240 on the Y-axis.

Draw a horizontal line to the curve, then drop a perpendicular to the X-axis.

Result: The upper quartile is approximately ₹ 10,250.

1. Median Wage: Approximately ₹ 9,300.
2. Employees below ₹ 8,500: Approximately 95 - 98 employees.
3. Senior Employees (> ₹ 11,500): Approximately 15 – 20 employees.
4. Upper Quartile (Q₃): Approximately ₹ 10,250.