Question

For the data given in the following table, find:

Class-mark	Frequency
35	8
40	10
45	12
50	15
55	20
60	15

1. mean
2. mode (using graph)
3. median (using grap)

Answer 1

Since the difference between two consecutive class-marks is 5, subtract `5/2​` = 2.5 from each class-mark to get the lower limit and add 2.5 to get the upper limit.

Class-mark (x)	Class interval	Frequency (f)	fx	Cumulative frequency
35	32.5 - 37.5	8	280	8
40	37.5 - 42.5	10	400	18
45	42.5 - 47.5	12	540	30
50	47.5 - 52.5	15	750	45
55	52.5 - 57.5	20	1100	65
60	57.5 - 62.5	15	900	80
Total	Σf = 80	Σfx = 3970

i. By formula,

Mean = `(Σfx)/(Σf) = 3970/80`​ = 49.625

Hence, mean = 49.625

ii. Calculating mode:

Steps of construction:

1. Draw a histogram of the given distribution.
2. Inside the highest rectangle, which represents the maximum frequency (or modal class) draw two lines AC and BD diagonally from the upper corners C and D of adjacent rectangles.
3. Through the point K (the point of intersection of diagonals AC and BD), draw KM perpendicular to the horizontal axis.
4. The value of point M on the horizontal axis represents the value of mode.

∴ Mode = 55 (approx).

Hence, mode = 55 (approx).

iii. n = Σf = 80

Median = `(n/2)^(th)` term = `80/2`​ = 40th term.

Steps of construction of ogive:

1. Take 1 cm = 5 units along x-axis.
2. Take 1 cm = 10 units along y-axis.
3. Ogive always starts from a point on x-axis representing the lower limit of the first class. Mark point (32.5, 0).
4. Take upper class limits along x-axis and corresponding cumulative frequencies along y-axis, mark the points (37.5, 8), (42.5, 18), (47.5, 30), (52.5, 45), (57.5, 65) and (62.5, 80).
5. Join the points marked by a free hand curve.
6. Mark 40 on y-axis, draw a horizontal line which meets the curve at point A.
7. Through point A, on the curve, draw a vertical line which meets x-axis at point B.

From graph,

B = 49.5 (approx)

Hence, median = 49.5 (approx).