Question

Find the mean and mode of the following data:

Class	15--20	20--25	25--30	30--35	35--40	40--45
Frequency	12	10	15	11	7	5

Hint:

For mean, use assumed mean method for easier calculation. For mode, identify the class with highest frequency and apply the mode formula.

Answer 1

Given Data:

Class	15–20	20–25	25–30	30–35	35–40	40–45
Frequency (f)	12	10	15	11	7	5

Step 1: Find the mid-point (x) of each class interval
Mid-point \( x = \frac{\text{Lower limit} + \text{Upper limit}}{2} \)
 

Class	Mid-point (x)
15 – 20	17.5
20 – 25	22.5
25 – 30	27.5
30 – 35	32.5
35 – 40	37.5
40 – 45	42.5

Step 2: Calculate \( f \times x \) for each class
 

Class	Frequency (f)	Mid-point (x)	f × x
15 – 20	12	17.5	210
20 – 25	10	22.5	225
25 – 30	15	27.5	412.5
30 – 35	11	32.5	357.5
35 – 40	7	37.5	262.5
40 – 45	5	42.5	212.5

Step 3: Find total frequency and total \( f \times x \)
Total frequency, \( \sum f = 12 + 10 + 15 + 11 + 7 + 5 = 60 \)
Total \( f \times x \), \( \sum f x = 210 + 225 + 412.5 + 357.5 + 262.5 + 212.5 = 1680 \)
Step 4: Calculate Mean
Mean \(= \frac{\sum f x}{\sum f} = \frac{1680}{60} = 28\)
Step 5: Find the Mode
Mode formula for grouped data:
Mode \(= l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h\)
Where:
- \( l \) = lower boundary of modal class
- \( f_1 \) = frequency of modal class
- \( f_0 \) = frequency of class before modal class
- \( f_2 \) = frequency of class after modal class
- \( h \) = class width
Step 6: Identify Modal Class
The modal class is the class with the highest frequency.
Frequencies: 12, 10, 15, 11, 7, 5
Highest frequency = 15, so modal class = 25 – 30
Step 7: Apply values in mode formula
- \( l = 25 \) (lower limit of modal class)
- \( f_1 = 15 \)
- \( f_0 = 10 \) (frequency before modal class)
- \( f_2 = 11 \) (frequency after modal class)
- \( h = 5 \) (class width)
Step 8: Calculate Mode
Mode \(= 25 + \left( \frac{15 - 10}{2 \times 15 - 10 - 11} \right) \times 5 = 25 + \left( \frac{5}{30 - 10 - 11} \right) \times 5 = 25 + \left( \frac{5}{9} \right) \times 5 = 25 + \frac{25}{9} = 25 + 2.78 = 27.78\)
Final Answer:
Mean = 28
Mode = 27.78