The average of a set of 15 observations is recorded, but later it is found that for one observation, the digit in the tens place was wrongly recorded as 8 instead of 3. After correcting the observation, the average is

Options

A. reduced by `(1)/(3)`
B. increased by `(10)/(3)`
C. reduced by `(10)/(3)`
D. reduced by 50

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript, while providing clear calculations and explanations.

### Step 1: Understand the average formula
The average (mean) of a set of observations is calculated using the formula:
\[ \text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
In this case, we have 15 observations.

### Step 2: Calculate the initial sum of observations
Let the average of the 15 observations be denoted as \( \bar{x} \). Therefore, the sum of the observations can be expressed as:
\[ \text{Sum of observations} = 15 \times \bar{x} \]

### Step 3: Identify the incorrect observation
One observation was incorrectly recorded. The digit in the tens place was recorded as 8 instead of 3. This means that the incorrect observation can be represented as:
\[ x_a = x + 80 \quad (\text{where } x \text{ is the original number}) \]
The correct observation should be:
\[ x_a' = x + 30 \]

### Step 4: Calculate the correction in the sum
To find the corrected sum of observations, we need to remove the incorrect observation and add the correct one:
\[ \text{Corrected Sum} = \text{Sum of observations} - x_a + x_a' \]
Substituting the values we have:
\[ \text{Corrected Sum} = 15 \bar{x} - (x + 80) + (x + 30) \]
This simplifies to:
\[ \text{Corrected Sum} = 15 \bar{x} - 80 + 30 = 15 \bar{x} - 50 \]

### Step 5: Calculate the new average
Now, we can find the new average using the corrected sum:
\[ \text{New Average} = \frac{\text{Corrected Sum}}{15} = \frac{15 \bar{x} - 50}{15} \]
This simplifies to:
\[ \text{New Average} = \bar{x} - \frac{50}{15} = \bar{x} - \frac{10}{3} \]

### Step 6: Conclusion
Thus, the new average after correcting the observation is:
\[ \text{New Average} = \bar{x} - \frac{10}{3} \]