Question

Find the mean of the following distribution :

Hint:

Using the Assumed Mean Method keeps the numbers small and manageable, reducing the chance of arithmetic errors compared to the Direct Method!

Answer 1

Step 1: Understanding the Question:
We are given a grouped frequency distribution table and need to compute its arithmetic mean.

Step 2: Key Formula or Approach:
We will use the Assumed Mean Method.
The formula for the mean is:
\[ \bar{x} = A + \frac{\sum f_i d_i}{\sum f_i} \cdot h \]
or directly:
\[ \bar{x} = A + \frac{\sum f_i d_i}{\sum f_i} \]
where \(A\) is the assumed mean and \(d_i = x_i - A\).

Step 3: Detailed Explanation:

• Determine the class marks (\(x_i = \frac{\text{Lower limit} + \text{Upper limit}}{2}\)) for each class:
- For 30–40: \(x_1 = 35\)
- For 40–50: \(x_2 = 45\)
- For 50–60: \(x_3 = 55\)
- For 60–70: \(x_4 = 65\)
- For 70–80: \(x_5 = 75\)

• Choose an Assumed Mean \(A\):
Let us choose \(A = 55\) (the class mark of the middle class).

• Construct a calculation table:

• Calculate the mean using the formula:
\[ \bar{x} = A + \frac{\sum f_i d_i}{\sum f_i} \]
\[ \bar{x} = 55 + \frac{90}{50} \]
\[ \bar{x} = 55 + 1.8 = 56.8 \]


Step 4: Final Answer:
The mean of the given distribution is 56.8.