Question

The following table gives the marks obtained by 50 students in a class test:

Marks	11 – 15	16 – 20	21 – 25	26 – 30	31 – 35	36 – 40	41 – 45	46 – 50
Number of students	2	3	6	7	14	12	4	2

Calculate the mean, median and mode for the above data.

Answer 1

1. Convert to continuous classes

Since the given classes are discontinuous (11 – 15, 16 – 20,...), we convert them into continuous class boundaries by subtracting 0.5 from each lower limit and adding 0.5 to each upper limit. We also determine the midpoints (x_i) and the cumulative frequencies (cf).

Marks (Given)	Class Boundaries	Frequency  (fi)	Midpoint (xi)	fi × xi
11 – 15	10.5 – 15.5	2	13	26
16 – 20	15.5 – 20.5	3	18	54
21 – 25	20.5 – 25.5	6	23	138
26 – 30	25.5 – 30.5	7	28	196
31 – 35	30.5 – 35.5	14	33	462
36 – 40	35.5 – 40.5	12	38	456
41 – 45	40.5 – 45.5	4	43	172
46 – 50	45.5 – 50.5	2	48	96
Total	–	Σfi = 50	–	Σfixi = 1600

2. Calculate the mean

The arithmetic mean is computed using the direct method formula:

Mean `(barx) = (sumf_ix_i)/(sumf_i)`

`barx = 1600/50 = 32`

3. Calculate the median

First, find the median class position:

`N/2 = 50/2 = 25`

The cumulative frequency just greater than or equal to 25 is 32, which corresponds to the continuous class interval 30.5 – 35.5.

Lower boundary of the median class (L) = 30.5.

Cumulative frequency of the preceding class (cf) = 18

Frequency of the median class (f) = 14

Class width (h) = 5

Now substitute these values into the median formula:

Median = `L + ((N/2 - cf)/f) xx h`

Median = `30.5 + ((25 - 18)/14) xx 5`

Median = `30.5 + (7/14) xx 5`

Median = 30.5 + 2.5

Median = 33

4. Calculate the mode

The highest frequency is 14, which makes 30.5 – 35.5 the modal class as well.

Lower boundary of the modal class (L) = 30.5

Frequency of the modal class (f₁) = 14

Frequency of the preceding class (f₀) = 7

Frequency of the succeeding class (f₂) = 12

Class width (h) = 5

Substitute these values into the mode formula:

Mode = `L + ((f_1 - f_0)/(2f_1 - f_0 - f_2)) xx h`

Mode = `30.5 + ((14 - 7)/(2(14) - 7 - 12)) xx 5`

Mode = `30.5 + (7/(28 - 19)) xx 5`

Mode = `30.5 + 35/9 ≈ 30.5 + 3.889`

Mode = 34.39