The following distribution represents the height of 160 students of a school.

Height (in cm)	No. of Students
140 - 145	12
145 - 150	20
150 - 155	30
155 - 160	38
160 - 165	24
165 - 170	16
170 - 175	12
175 - 180	8

Draw an ogive for the given distribution taking 2 cm = 5 cm of height on one axis and 2 cm = 20 students on the other axis. Using the graph, determine :

(i) the median height.

(ii) the inter-quartile range.

(iii) the number of students whose height is above 172 cm ?

(i) Cumulative frequency distribution table :

Here, n = 160, which is even.

Median = $\dfrac{n}{2}$ th term = $\dfrac{160}{2}$ = 80th term.

Steps of construction :

1. Take 1 cm = 5 cm on x-axis.

2. Take 1 cm = 20 students on y-axis.

3. Since, x axis starts at 140 hence, a kink is drawn at the starting of x-axis. Plot the point (140, 0) as ogive starts on x-axis representing lower limit of first class.

4. Plot the points (145, 12), (150, 32), (155, 62), (160, 100), (165, 124), (170, 140), (175, 152) and (180, 160).

5. Join the points by a free-hand curve.

6. Draw a line parallel to x-axis from point J (no. of students) = 80, touching the graph at point K. From point K draw a line parallel to y-axis touching x-axis at point L.

From graph, L = 157.5

Hence, median = 157.5 cm

(ii) Here, n = 160, which is even.

By formula,

Lower quartile = $\dfrac{n}{4} = \dfrac{160}{4}$ = 40th term.

Draw a line parallel to x-axis from point P (no. of students) = 40, touching the graph at point Q. From point Q draw a line parallel to y-axis touching x-axis at point R.

From graph, R = 152

Upper quartile = $\dfrac{3n}{4} = \dfrac{3 \times 160}{4}$ = 120th term.

Draw a line parallel to x-axis from point M (no. of students) = 120, touching the graph at point N. From point N draw a line parallel to y-axis touching x-axis at point O.

From graph, O = 164

Inter quartile range = Upper quartile - Lower quartile

= 164 - 152 = 12.

Hence, inter quartile range = 12.

(iii) Draw a line parallel to y-axis from point S (height) = 172 cm, touching the graph at point T. From point T draw a line parallel to x-axis touching y-axis at point U.

From graph, U = 144.

∴ 144 students have height less than or equal to 172 cm.

No. of students whose height is more than 172 cm = 160 - 144 = 16.

Hence, no. of students whose height is more than 172 cm = 16.