The table shows the distribution of the scores obtained by 160 shooters in a shooting competition. Use a graph sheet and draw an ogive for the distribution (take 2 cm = 10 scores on the x-axis and 2 cm = 20 shooters on the y-axis.)

Scores	Number of shooters
0 - 10	9
10 - 20	13
20 - 30	20
30 - 40	26
40 - 50	30
50 - 60	22
60 - 70	15
70 - 80	10
80 - 90	8
90 - 100	7

Use your graph to estimate the following:

(i) the median

(ii) the inter-quartile range

(iii) the number of shooters who obtained a score of more than 85%

Cumulative frequency distribution table :

Here, n = 160, which is even.

(i) Steps of construction:

1. Take 1 cm along x-axis = 10 scores

2. Take 2 cm along y-axis = 20 shooters

3. Plot the point (0, 0) as ogive starts from x- axis representing lower limit of first class.

4. Plot the points (10, 9), (20, 22), (30, 42), (40, 68), (50, 98), (60, 120), (70, 135), (80, 145), (90, 153), (100, 160)

5. Joint the points by a free hand curve.

To find the median :

Let A be the point on y-axis representing frequency = $\dfrac{\text{n}}{2} = \dfrac{160}{2}$ = 80.

Through A draw a horizontal line to meet the ogive at P. Through P, draw a vertical line to meet the x-axis at M. The abscissa of the points M represents 44.

Hence, the median score is 44.

(ii) To find lower quartile:

Let B be the point on y-axis representing frequency = $\dfrac{\text{n}}{4} = \dfrac{160}{4}$ = 40.

Through B, draw a horizontal line to meet the ogive at Q. Through Q, draw a vertical line to meet the x-axis at N. The abscissa of the point N represents 29.

To find upper quartile:

Let C be the point on y-axis representing frequency = $\dfrac{3\text{n}}{4} = \dfrac{3 \times 160}{4} = \dfrac{480}{4}$ = 120.

Through C, draw a horizontal line to meet the ogive at R. Through R, draw a vertical line to meet the x-axis at S. The abscissa of the point S represents 60.

Inter-quartile range = Upper quartile - Lower quartile = 60 - 29 = 31.

Hence, the inter quartile range is = 31.

(iii) Total score = 100

So, more than 85% score mean more than 85 score.

Let T be the point on x-axis representing scores = 85

Through T, draw a vertical line to meet the ogive at E. Through E, draw a horizontal line to meet the y-axis at D. The ordinate of the point D represents 149.

Shooters who have scored less than 85% = 149

So, students scoring more than 85% = Total students - Students who have scored less = 160 - 149 = 11.

Hence, there are 11 number of shooters who obtained more than 85% score.