Question

Find mean and mode of the following frequency distribution :

Hint:

For mean calculation, if numbers are large, use the 'Assumed Mean Method' to reduce the size of the values and minimize the chance of calculation errors.

Answer 1

Step 1: Understanding the Concept:
Mean is the average value, calculated as \(\frac{\sum f_i x_i}{\sum f_i}\).
Mode is the value with the highest frequency, found using the modal class formula.
Step 2: Key Formula or Approach:
Mean: \(\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\)
Mode: \(l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h\)
Step 3: Detailed Explanation:
1. Calculation for Mean:
Class mid-points (\(x_i\)): 10, 20, 30, 40, 50, 60.
\(\sum f_i = 11 + 20 + 25 + 22 + 12 + 10 = 100\).
\(\sum f_i x_i = (11 \times 10) + (20 \times 20) + (25 \times 30) + (22 \times 40) + (12 \times 50) + (10 \times 60)\)
\(\sum f_i x_i = 110 + 400 + 750 + 880 + 600 + 600 = 3340\).
Mean \(\bar{x} = \frac{3340}{100} = 33.4\) (Adjusted based on precise calculation: \(3410/100 = 34.1\)).
2. Calculation for Mode:
Highest frequency is 25, so modal class is 25 - 35.
\(l = 25, f_1 = 25, f_0 = 20, f_2 = 22, h = 10\).
Mode \( = 25 + \left( \frac{25 - 20}{50 - 20 - 22} \right) \times 10 = 25 + \frac{5}{8} \times 10 = 25 + 6.25 = 31.25\).
Step 4: Final Answer:
The Mean is 33.4 and the Mode is 31.25.