For the data given in the following tables, find :

(i) mean

(ii) mode (using graph)

(iii) median (using graph)

(a)

Class-mark	Frequency
15	10
25	12
35	14
45	16
55	8

(b)

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

(a) Since the difference between two consecutive class-marks is 10, subtract $\dfrac{10}{2}$ = 5 from each class-mark to get the lower limit and add 5 to get the upper limit.

(i) By formula,

Mean = $\dfrac{Σfx}{Σf} = \dfrac{2100}{60}$ = 35.

Hence, mean = 35.

(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 = 42 (approx).

Hence, mode = 42 (approx).

(iii) n = Σf = 60

Median = $\Big(\dfrac{n}{2}\Big)^{th}$ term = $\dfrac{60}{2}$ = 30th term.

Steps of construction of ogive :

1. Take 1 cm = 10 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 (10, 0).

4. Take upper class limits along x-axis and corresponding cumulative frequencies along y-axis, mark the points (20, 10), (30, 22), (40, 36), (50, 52) and (60, 60).

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

6. Mark 30 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 = 36 (approx)

Hence, median = 36 (approx).

(b) Since the difference between two consecutive class-marks is 5, subtract $\dfrac{5}{2}$ = 2.5 from each class-mark to get the lower limit and add 2.5 to get the upper limit.

(i) By formula,

Mean = $\dfrac{Σfx}{Σf} = \dfrac{3970}{80}$ = 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 = $\Big(\dfrac{n}{2}\Big)^{th}$ term = $\dfrac{80}{2}$ = 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).