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

Mean Calculation:

The mean of grouped data is given by the formula:

\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]

Calculate midpoints \( x_i \) of each class:

Class Interval	\( f_i \)	\( x_i \)	\( f_i x_i \)
15--20	12	17.5	210.0
20--25	10	22.5	225.0
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
Total	60	—	1680.0

\[ \bar{x} = \frac{1680.0}{60} = 28.0 \]

Mean = 28.0

Mode Calculation:

Modal class is 25--30 (highest frequency = 15)

Use the mode formula:

\[ \text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \]

Where:

• \( L = 25 \)
• \( f_1 = 15 \)
• \( f_0 = 10 \)
• \( f_2 = 11 \)
• \( h = 5 \)

\[ \text{Mode} = 25 + \left( \frac{15 - 10}{2 \times 15 - 10 - 11} \right) \times 5 = 25 + \left( \frac{5}{9} \right) \times 5 \]

\[ \text{Mode} = 25 + \frac{25}{9} \approx 25 + 2.78 = 27.78 \]

Mode ≈ 27.78

Final Answers:

• Mean = 28.0
• Mode ≈ 27.78