Mean marks of 50 students were found to be 78.4. But later it was detected that 95 was misread as 59 and 25 was misread as 52. What is the difference between correct mean and incorrect mean?
Understanding the Problem: Calculating the Correct Mean
The question asks us to find the difference between the correct mean and the incorrect mean of marks for 50 students. We are given the initial incorrect mean and information about the errors made when recording two student marks.
Here's what we know:
- Number of students (n) = 50
- Incorrect Mean = 78.4
- A mark of 95 was misread as 59.
- A mark of 25 was misread as 52.
Step-by-Step Solution for Mean Correction
1. Calculate the Incorrect Total Sum of Marks
The mean is calculated by dividing the total sum of observations by the number of observations. We can use the given incorrect mean and the number of students to find the incorrect total sum of marks.
Formula:
\(\text{Mean} = \frac{\text{Total Sum}}{\text{Number of Observations}}\)
So, the Incorrect Total Sum is:
\(\text{Incorrect Sum} = \text{Incorrect Mean} \times \text{Number of Students}\)
\(\text{Incorrect Sum} = 78.4 \times 50\)
\(\text{Incorrect Sum} = 3920\)
The total sum calculated with the errors was 3920.
2. Identify the Impact of the Errors
Two marks were recorded incorrectly. To find the correct total sum, we need to remove the incorrect values that were included and add the correct values that should have been included.
- Incorrect value 1: 59 (should have been 95)
- Incorrect value 2: 52 (should have been 25)
The sum of the incorrect values is \(59 + 52 = 111\).
The sum of the correct values should have been \(95 + 25 = 120\).
The net change required in the total sum is the sum of correct values minus the sum of incorrect values: \(120 - 111 = 9\). This means the incorrect sum was 9 less than the correct sum.
3. Calculate the Correct Total Sum of Marks
We can find the correct total sum by adjusting the incorrect total sum based on the errors.
\(\text{Correct Sum} = \text{Incorrect Sum} - (\text{Sum of incorrect values}) + (\text{Sum of correct values})\)
or
\(\text{Correct Sum} = \text{Incorrect Sum} + (\text{Sum of correct values} - \text{Sum of incorrect values})\)
\(\text{Correct Sum} = 3920 - 111 + 120\)
\(\text{Correct Sum} = 3920 + 9\)
\(\text{Correct Sum} = 3929\)
The correct total sum of marks is 3929.
4. Calculate the Correct Mean
Now we can calculate the correct mean using the correct total sum and the number of students.
\(\text{Correct Mean} = \frac{\text{Correct Sum}}{\text{Number of Students}}\)
\(\text{Correct Mean} = \frac{3929}{50}\)
\(\text{Correct Mean} = 78.58\)
The correct mean of the marks is 78.58.
5. Calculate the Difference Between Correct and Incorrect Mean
Finally, we need to find the difference between the correct mean and the incorrect mean.
\(\text{Difference} = \text{Correct Mean} - \text{Incorrect Mean}\)
\(\text{Difference} = 78.58 - 78.4\)
\(\text{Difference} = 0.18\)
The difference between the correct mean and the incorrect mean is 0.18.
| Item | Value |
|---|---|
| Number of Students (n) | 50 |
| Incorrect Mean | 78.4 |
| Incorrect Sum | 3920 |
| Incorrectly Read Values | 59, 52 |
| Correct Values | 95, 25 |
| Sum of Incorrect Values | 111 |
| Sum of Correct Values | 120 |
| Change in Sum | +9 |
| Correct Sum | 3929 |
| Correct Mean | 78.58 |
| Difference (Correct - Incorrect) | 0.18 |
Revision Table: Key Concepts
| Concept | Definition/Formula | Relevance to Problem |
|---|---|---|
| Mean (Average) | Sum of observations / Number of observations | Central measure of data; needs correction |
| Total Sum | Mean × Number of observations | Intermediate step to find correct values |
| Error in Data | Incorrect recording of values | Source of discrepancy in mean |
| Data Correction | Adjusting sum by removing incorrect values and adding correct ones | Method used to find correct sum and mean |
Additional Information: Understanding Mean and Errors
The mean, or average, is a fundamental measure of central tendency in statistics. It gives us a single value that represents the typical value in a dataset. However, the mean is sensitive to errors in the data. If even one value is recorded incorrectly, it can affect the calculated mean.
When errors are detected in a dataset after the mean has been calculated, it's necessary to correct the total sum first. The process involves subtracting the values that were wrongly included and adding the values that should have been included. This adjusted total sum is then used with the original number of observations to calculate the correct mean.
This type of correction is important in various fields, such as surveying data, financial reporting, or academic grading, to ensure that the calculated statistics accurately reflect the true underlying data.
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