The average height of 22 students of a class is 140 cm and the average height of 28 students of another class is 152 cm. What is the average height of students of both classes?
146. 72 cm
Calculating Combined Average Height of Students
This problem asks us to find the average height of students when two different classes with different numbers of students and different average heights are combined. To solve this, we need to understand the concept of average and how to calculate a weighted average.
Understanding Average
The average (or mean) of a set of values is calculated by summing all the values and dividing by the total number of values. The formula is:
\( \text{Average} = \frac{\text{Sum of values}}{\text{Number of values}} \)
From this, we can also say that:
\( \text{Sum of values} = \text{Average} \times \text{Number of values} \)
Calculating Total Height for Each Class
We are given the average height and the number of students for each class. We can use the second formula above to find the total height of students in each class.
- Class 1:
- Number of students = 22
- Average height = 140 cm
- Total height of Class 1 students = \( 22 \times 140 \text{ cm} \)
- Total height of Class 1 students = 3080 cm
- Class 2:
- Number of students = 28
- Average height = 152 cm
- Total height of Class 2 students = \( 28 \times 152 \text{ cm} \)
- Total height of Class 2 students = 4256 cm
Calculating Total Number of Students and Total Height
To find the average height of students of both classes combined, we need the total number of students and the total height of all students from both classes.
- Total number of students = Number of students in Class 1 + Number of students in Class 2
- Total number of students = \( 22 + 28 = 50 \)
- Total height of all students = Total height of Class 1 students + Total height of Class 2 students
- Total height of all students = \( 3080 \text{ cm} + 4256 \text{ cm} = 7336 \text{ cm} \)
Calculating the Combined Average Height
Now we can use the average formula for the combined group:
\( \text{Combined Average Height} = \frac{\text{Total height of all students}}{\text{Total number of students}} \)
\( \text{Combined Average Height} = \frac{7336 \text{ cm}}{50} \)
Let's perform the division:
\( \frac{7336}{50} = \frac{733.6}{5} \)
\( \frac{733.6}{5} = 146.72 \)
So, the average height of students of both classes combined is 146.72 cm.
| Class | Number of Students (\(n\)) | Average Height (cm) (\(A\)) | Total Height (cm) (\(n \times A\)) |
|---|---|---|---|
| Class 1 | 22 | 140 | \(22 \times 140 = 3080\) |
| Class 2 | 28 | 152 | \(28 \times 152 = 4256\) |
| Total | \(22 + 28 = 50\) | - | \(3080 + 4256 = 7336\) |
Combined Average Height = \( \frac{7336}{50} = 146.72 \text{ cm} \)
Summary of Calculation Steps
- Calculate the total height for the first class: multiply the number of students by their average height (\(22 \times 140\)).
- Calculate the total height for the second class: multiply the number of students by their average height (\(28 \times 152\)).
- Find the total height of all students: add the total heights from both classes (\(3080 + 4256\)).
- Find the total number of students: add the number of students from both classes (\(22 + 28\)).
- Calculate the combined average height: divide the total height by the total number of students (\(7336 \div 50\)).
The calculated average height of the students of both classes is 146.72 cm.
Revision Table: Average Height Calculation
| Concept | Formula | Application |
| Average | Sum of values / Number of values | Used to define the central tendency of a dataset. |
| Total Sum (from Average) | Average × Number of values | Used to find the total sum when average and count are known. Essential for combining groups. |
| Combined Average | (Sum of Group 1 + Sum of Group 2) / (Count of Group 1 + Count of Group 2) | Used to find the average of a larger group formed by combining smaller groups with known averages and counts. Also called weighted average. |
Additional Information: Weighted Average Explained
The combined average calculated here is a specific type of average called a weighted average. It is used when different groups contribute differently to the overall average based on their size. In this case, the average height of each class is 'weighted' by the number of students in that class. Since Class 2 has more students (28) than Class 1 (22), its average height (152 cm) has a greater influence on the final combined average than the average height of Class 1 (140 cm). This is why the combined average (146.72 cm) is closer to 152 cm than to 140 cm.
The formula for weighted average with two groups is:
\( \text{Weighted Average} = \frac{(n_1 \times A_1) + (n_2 \times A_2)}{n_1 + n_2} \)
Where:
- \(n_1\) = number of items in group 1
- \(A_1\) = average of group 1
- \(n_2\) = number of items in group 2
- \(A_2\) = average of group 2
This is exactly the method we used by first calculating the total height (\(n_1 \times A_1\) and \(n_2 \times A_2\)) and then dividing by the total number of students (\(n_1 + n_2\)). Understanding weighted averages is crucial for solving problems involving averages of combined groups.
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