The average weight of a team of 20 people was calculated to be 59.8 kg and it was later discovered that one weight was misread as 68 kg instead of 77 kg. The correct average weight is:
60.25 kg
Calculating Correct Average Weight of a Team
The problem asks us to find the correct average weight of a team after discovering an error in one of the recorded weights. Initially, the average weight of 20 people was calculated based on a misread value.
Understanding the Average Weight Calculation
The average weight of a group of people is calculated by dividing the total weight of all individuals by the number of people in the group.
Average Weight = \( \frac{\text{Total Weight}}{\text{Number of People}} \)
Step-by-Step Solution for Average Weight Correction
Step 1: Calculate the initial total weight based on the incorrect average.
We are given the initial average weight and the number of people. We can use these values to find the total weight that was calculated initially.
- Number of people = 20
- Initial average weight = 59.8 kg
Initial Total Weight = Initial Average Weight \(\times\) Number of People
Initial Total Weight = \( 59.8 \text{ kg} \times 20 \)
Initial Total Weight = \( 1196 \text{ kg} \)
This 1196 kg is the sum of the weights including the incorrect value of 68 kg.
Step 2: Determine the difference between the correct and incorrect weights.
One weight was misread as 68 kg instead of the correct value of 77 kg. To find the correct total weight, we need to adjust for this error. The difference between the correct and incorrect value tells us how much the total sum was off.
- Correct weight = 77 kg
- Incorrect weight = 68 kg
Difference = Correct Weight \(- \) Incorrect Weight
Difference = \( 77 \text{ kg} - 68 \text{ kg} \)
Difference = \( 9 \text{ kg} \)
The correct weight is 9 kg higher than the misread weight.
Step 3: Calculate the correct total weight.
Since the correct weight is 9 kg more than the weight that was used in the initial calculation, the correct total weight will be 9 kg more than the initial total weight.
Correct Total Weight = Initial Total Weight \(+\) Difference
Correct Total Weight = \( 1196 \text{ kg} + 9 \text{ kg} \)
Correct Total Weight = \( 1205 \text{ kg} \)
This is the true sum of the weights of the 20 people.
Step 4: Calculate the correct average weight.
Now that we have the correct total weight and the number of people, we can calculate the correct average weight.
Correct Average Weight = \( \frac{\text{Correct Total Weight}}{\text{Number of People}} \)
Correct Average Weight = \( \frac{1205 \text{ kg}}{20} \)
Correct Average Weight = \( 60.25 \text{ kg} \)
Therefore, the correct average weight of the team is 60.25 kg.
| Calculation Step | Formula / Operation | Value |
|---|---|---|
| Initial Total Weight | Average \(\times\) Count | \(59.8 \times 20 = 1196 \text{ kg}\) |
| Weight Difference | Correct \(- \) Incorrect | \(77 - 68 = 9 \text{ kg}\) |
| Correct Total Weight | Initial Total \(+\) Difference | \(1196 + 9 = 1205 \text{ kg}\) |
| Correct Average Weight | Correct Total \(\div\) Count | \(\frac{1205}{20} = 60.25 \text{ kg}\) |
Revision Table: Key Concepts in Average Calculations
When dealing with average calculations and errors, remember these points:
- Average is Total Sum divided by the Number of items.
- Errors in individual values affect the Total Sum, and consequently the Average.
- To correct the average, first correct the Total Sum by adding or subtracting the difference between the correct and incorrect values.
- Then, recalculate the average using the corrected Total Sum and the original Number of items.
Additional Information: Average and Mean
The terms "average" and "mean" are often used interchangeably, especially in the context of the arithmetic mean. The arithmetic mean is the sum of a set of values divided by the number of values in the set. This is precisely what we calculated here for the weight of the team members. Understanding how errors affect the sum is crucial for correcting averages in data sets.
Other types of means exist, such as the geometric mean and harmonic mean, but the average weight calculation in this problem specifically refers to the arithmetic mean.
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