Here's the step-by-step solution to find the correct mean:
1. Initial Mean and Total Sum:
- The initial mean of the 200 items was 50.
- Therefore, the initial total sum of all the items is:
[tex]\( \text{Initial Total Sum} = \text{Mean} \times \text{Number of Items} \)[/tex]
[tex]\[ \text{Initial Total Sum} = 50 \times 200 \][/tex]
[tex]\[ \text{Initial Total Sum} = 10000 \][/tex]
2. Sum of Wrong Values:
- The two items were initially taken as 92 and 8.
- Therefore, their sum is:
[tex]\[ \text{Sum of Wrong Values} = 92 + 8 \][/tex]
[tex]\[ \text{Sum of Wrong Values} = 100 \][/tex]
3. Sum of Correct Values:
- The correct values should have been 192 and 88.
- Therefore, their sum is:
[tex]\[ \text{Sum of Correct Values} = 192 + 88 \][/tex]
[tex]\[ \text{Sum of Correct Values} = 280 \][/tex]
4. Difference Due to Correction:
- The difference between the correct sum and the wrong sum is:
[tex]\[ \text{Difference} = \text{Sum of Correct Values} - \text{Sum of Wrong Values} \][/tex]
[tex]\[ \text{Difference} = 280 - 100 \][/tex]
[tex]\[ \text{Difference} = 180 \][/tex]
5. Corrected Total Sum:
- To get the corrected total sum, add the difference to the initial total sum:
[tex]\[ \text{Corrected Total Sum} = \text{Initial Total Sum} + \text{Difference} \][/tex]
[tex]\[ \text{Corrected Total Sum} = 10000 + 180 \][/tex]
[tex]\[ \text{Corrected Total Sum} = 10180 \][/tex]
6. Corrected Mean:
- Finally, the corrected mean is the corrected total sum divided by the number of items:
[tex]\[ \text{Corrected Mean} = \frac{\text{Corrected Total Sum}}{\text{Number of Items}} \][/tex]
[tex]\[ \text{Corrected Mean} = \frac{10180}{200} \][/tex]
[tex]\[ \text{Corrected Mean} = 50.9 \][/tex]
Therefore, the correct mean is 50.9.
