If the average of p numbers is q^2 and that of q numbers is p^2, then the average of (p + q) numbers is:
Understand the Problem
The question is asking to find the average of two sets of numbers given their respective averages. It requires an understanding of how to compute the overall average from two distinct groups.
Answer
The overall average is given by: $$ A_{overall} = \frac{n_1 \cdot A_1 + n_2 \cdot A_2}{n_1 + n_2} $$
Answer for screen readers
The overall average is given by the formula:
$$ A_{overall} = \frac{n_1 \cdot A_1 + n_2 \cdot A_2}{n_1 + n_2} $$
Steps to Solve
- Identify Given Data
Let the first set have $n_1$ numbers with an average of $A_1$. The sum of this set is $S_1 = n_1 \cdot A_1$.
Let the second set have $n_2$ numbers with an average of $A_2$. The sum of this set is $S_2 = n_2 \cdot A_2$.
- Calculate Total Sum and Total Numbers
The total sum of both sets is:
$$ S_{total} = S_1 + S_2 = n_1 \cdot A_1 + n_2 \cdot A_2 $$
The total number of elements in both sets is:
$$ n_{total} = n_1 + n_2 $$
- Calculate Overall Average
We find the overall average, $A_{overall}$, using the total sum and total numbers:
$$ A_{overall} = \frac{S_{total}}{n_{total}} = \frac{n_1 \cdot A_1 + n_2 \cdot A_2}{n_1 + n_2} $$
The overall average is given by the formula:
$$ A_{overall} = \frac{n_1 \cdot A_1 + n_2 \cdot A_2}{n_1 + n_2} $$
More Information
This calculation is useful in statistics to find a combined average when dealing with multiple data sets.
For example, if set one consists of 5 students with an average score of 80 and set two consists of 10 students with an average score of 60, we can use this method to find the overall average score of all students.
Tips
- Forgetting to multiply the averages by the number of elements to find the total sum for each group.
- Confusing the total average formula, leading to incorrect data handling.
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