Combined Distribution Mean Calculation

This problem involves finding the mean of a combined distribution formed from several smaller groups (components). Each group has a specific frequency (number of items) and its own mean. To find the overall mean of all components combined, we use a weighted average approach, where the frequencies act as weights.

Understanding the Combined Mean Formula

The formula for the mean of a combined distribution is:

\(\bar{X} = \frac{\sum_{i=1}^{k} (n_i \bar{x}_i)}{\sum_{i=1}^{k} n_i}\)

Where:

• \(\bar{X}\) is the mean of the combined distribution.
• \(k\) is the number of components (in this case, 3).
• \(n_i\) is the frequency of the i-th component.
• \(\bar{x}_i\) is the mean of the i-th component.
• \(n_i \bar{x}_i\) is the sum of values for the i-th component.
• \(\sum_{i=1}^{k} n_i\) is the total frequency of the combined distribution.

Data from the Question

Let's list the information given for each component:

Step-by-Step Calculation

1. Calculate the Total Frequency (\(N\))

Sum the frequencies of all components:

\(N = n_1 + n_2 + n_3\)

\(N = 45 + 40 + 55\)

\(N = 140\)

2. Calculate the Sum of (Frequency × Mean) for Each Component

Multiply the frequency by the mean for each component:

• Component 1: \(n_1 \bar{x}_1 = 45 \times 2 = 90\)
• Component 2: \(n_2 \bar{x}_2 = 40 \times 2.5 = 100\)
• Component 3: \(n_3 \bar{x}_3 = 55 \times 2 = 110\)

3. Calculate the Sum of All Products

Add the results from step 2:

\(\sum (n_i \bar{x}_i) = 90 + 100 + 110\)

\(\sum (n_i \bar{x}_i) = 300\)

4. Calculate the Combined Mean (\(\bar{X}\))

Apply the combined mean formula using the results from steps 1 and 3:

\(\bar{X} = \frac{\sum (n_i \bar{x}_i)}{N}\)

\(\bar{X} = \frac{300}{140}\)

Simplify the fraction:

\(\bar{X} = \frac{30}{14} = \frac{15}{7}\)

Convert the fraction to a decimal:

\(\bar{X} \approx 2.142857...\)

Conclusion

Rounding the result to two decimal places, the mean of the combined distribution is approximately 2.14.