1. What is the formula for calculating the Mean (Direct Method)?

The formula is:
Mean (\(\bar{x}\)) = \(\frac{\sum f_i x_i}{\sum f_i}\)

2. How do you calculate the Mean using the Assumed Mean Method?

The formula is:
\(\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i}\), where \(d_i = x_i – A\).

3. What is the formula for Median of grouped data?

The formula is:
Median = \(l + \left(\frac{\frac{N}{2} – \text{CF}}{f}\right) \times h\)

Where:

• \(l\): Lower boundary of the median class
• \(N\): Total frequency (\(\sum f_i\))
• \(\text{CF}\): Cumulative frequency of the class preceding the median class
• \(f\): Frequency of the median class
• \(h\): Class width

4. How is Mode calculated for grouped data?

The formula is:
Mode = \(l + \left(\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\right) \times h\)

Where:

• \(f_1\): Frequency of the modal class
• \(f_0\): Frequency of the class before the modal class
• \(f_2\): Frequency of the class after the modal class
• \(h\): Class width

5. How do you solve the given problem to calculate the Mean using the Assumed Mean Method?

Class Intervals: 0-10, 10-20, 20-30, 30-40

Frequency: 5, 8, 15, 12

Assumed Mean: \(A = 25\)

Solution:

1. Calculate midpoints \(x_i\) for each class: 5, 15, 25, 35.
2. Find \(d_i = x_i – A\): -20, -10, 0, 10.
3. Multiply \(f_i\) and \(d_i\): -100, -80, 0, 120.
4. Sum \(f_i\): 40. Sum \(f_i d_i\): -60.
5. Use formula: \(\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i} = 25 + \frac{-60}{40} = 23.5\).

Mean: 23.5

6. How do you find the Median for the given data?

Class Intervals: 0-10, 10-20, 20-30, 30-40

Frequency: 5, 8, 15, 12

Total Frequency (\(N\)): 40

Median Class: 20-30

Solution:

1. \(l = 20, h = 10, \text{CF} = 13, f = 15, N = 40\).
2. Use formula: \(\text{Median} = l + \left(\frac{\frac{N}{2} – \text{CF}}{f}\right) \times h\).
3. \(\text{Median} = 20 + \left(\frac{20 – 13}{15}\right) \times 10 = 20 + 4.67 = 24.67\).

Median: 24.67

7. How do you determine the Mode for the given data?

Class Intervals: 0-10, 10-20, 20-30, 30-40

Frequency: 5, 8, 15, 12

Modal Class: 20-30

Solution:

1. \(l = 20, h = 10, f_1 = 15, f_0 = 8, f_2 = 12\).
2. Use formula: \(\text{Mode} = l + \left(\frac{f_1 – f_0}{2f_1 – f_0 – f_2}\right) \times h\).
3. \(\text{Mode} = 20 + \left(\frac{15 – 8}{30 – 8 – 12}\right) \times 10 = 20 + 4.38 = 24.38\).

Mode: 24.38