Question

A student while calculating the mean of 25 observations obtained a mean of 56 cm. It was later discovered at the time of checking that he had wrongly copied down an observation as 64. What is the mean if the correct value is omitted?

• A. 55
• B. 57
• C. 56.33
• D. 55.67

Hint:

When an outlier or incorrect observation is omitted, remember to decrease the denominator (\( n \)) by \( 1 \) along with subtracting the value from the numerator.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem requires calculating the corrected arithmetic mean of a dataset after removing an incorrect observation from the calculation.
Key Formula or Approach:
The arithmetic mean is given by:
\[ \bar{X} = \frac{\sum X}{n} \implies \sum X = n \cdot \bar{X} \] When an observation is omitted entirely, both the sum of the observations (\( \sum X \)) and the total count of observations (\( n \)) decrease.

Step 2: Detailed Explanation:
Let us perform the calculations step-by-step:
- Initial State:
Number of observations, \( n_{\text{old}} = 25 \)
Incorrect mean, \( \bar{X}_{\text{old}} = 56 \)
Incorrect sum of observations:
\[ \sum X_{\text{old}} = 25 \times 56 = 1400 \] - Applying the Correction:
The incorrect observation wrongly copied down was \( 64 \).
We are instructed to find the new mean if this observation is completely omitted (deleted) from the dataset.
The new sum of the remaining observations is:
\[ \sum X_{\text{new}} = \sum X_{\text{old}} - 64 \] \[ \sum X_{\text{new}} = 1400 - 64 = 1336 \] - Adjusting the Count:
Since one observation is removed, the new number of observations is:
\[ n_{\text{new}} = 25 - 1 = 24 \] - Calculating the New Mean:
\[ \bar{X}_{\text{new}} = \frac{\sum X_{\text{new}}}{n_{\text{new}}} = \frac{1336}{24} \approx 55.6667 \] Rounding to two decimal places gives \( 55.67 \).

Step 3: Final Answer:
The mean of the remaining observations is \( 55.67 \).
Therefore, the correct choice is Option (D).