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:
| Component | Frequency (\(n_i\)) | Mean (\(\bar{x}_i\)) |
|---|---|---|
| 1 | 45 | 2 |
| 2 | 40 | 2.5 |
| 3 | 55 | 2 |
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.
Similar Questions
What is the median of the distribution ?
- What is the mode of the distribution?
- Consider the following data :
110, 41, 43, 95, 127, 99, 61, 92, 71, 93, 110, 36.
If 93 is replaced by 94, then consider the following statements:
1. The difference between new median and old median is 1.
2. The difference between new mean and old mean is less than 0·1.
3. The difference between new mode and old mode is zero.
Which of the statements given above are correct? - Which measure of central tendency is least affected by the presence of extreme observations in the data?
- To find the average ratio like price/unit, work done/hour, kilometre/hour under certain conditions, the suitable measure of central tendency applicable is
- The frequency distribution of marks of 100 candidates in a particular examination is as follows:What are the average marks of the candidates?
Marks Number of Candidates More than 10 100 More than 20 75 More than 30 60 More than 40 40 - The arithmetic mean of 200 observations is 60. If 5 is multiplied to each observation, then what will be the new arithmetic mean?
- Which one of the following is a positional average?
- What is the value of \(u\)?
- What is the value of \(v\)?
Important Questions from Elementary Statistics
In a colony 5 families have 1 child, 7 families have 2 children, 8 families have 3 children and 3 families have 4 children.What is the mode of the number of children.
For which set of numbers do the mean, median and mode all have the same value?
The median of 5, 8, 25, 22, 34, 18 is
Find the mean of first 5 two-digit multiples of 4.
The difference between the upper limit and the lower limit of a class is called :
