Question

Find the ‘mean’ and ‘mode’ marks of the following data:

Hint:

For grouped data, use class midpoints to calculate mean and identify the modal class (highest frequency) to find mode.

Answer 1

Given Data:

Marks	Number of Students (f)	Mid-point (x)	f × x
0 – 5	2	2.5	5
5 – 10	3	7.5	22.5
10 – 15	8	12.5	100
15 – 20	15	17.5	262.5
20 – 25	14	22.5	315
25 – 30	8	27.5	220

Step 1: Calculate total frequency and total \(f \times x\)
\[ \sum f = 2 + 3 + 8 + 15 + 14 + 8 = 50 \] \[ \sum f x = 5 + 22.5 + 100 + 262.5 + 315 + 220 = 925 \]

Step 2: Calculate Mean
\[ \text{Mean} = \frac{\sum f x}{\sum f} = \frac{925}{50} = 18.5 \]

Step 3: Find Modal class
Modal class is the class interval with highest frequency.
Frequencies: 2, 3, 8, 15, 14, 8
Highest frequency = 15 (class 15 – 20)
So, modal class = 15 – 20

Step 4: Calculate Mode using formula for grouped data
\[ \text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h \] Where:
\(l = 15\) (lower boundary of modal class)
\(f_1 = 15\) (frequency of modal class)
\(f_0 = 8\) (frequency of class before modal class)
\(f_2 = 14\) (frequency of class after modal class)
\(h = 5\) (class width)

Substitute values:
\[ \text{Mode} = 15 + \frac{15 - 8}{2 \times 15 - 8 - 14} \times 5 = 15 + \frac{7}{30 - 8 - 14} \times 5 = 15 + \frac{7}{8} \times 5 = 15 + 4.375 = 19.375 \]

Final Answer:
Mean marks = 18.5
Mode marks = 19.375