The values of measurement of a physical quantity in five trails were found to be ` 1.51 , 1.53 , 1.53, 1.52 , and 1.54`.

Options

A. Average absolute error is `0.01`.
B. Relative error is `0.01.`
C. Percentage error is `0.01%`.
D. Percentage error is `1%`.

Explanation

To solve the problem step by step, we will find the mean, absolute error, relative error, and percentage error based on the given measurements.

### Step 1: Calculate the Mean
The mean (average) of the measurements can be calculated using the formula:

\[ \text{Mean} = \frac{a_1 + a_2 + a_3 + a_4 + a_5}{5} \]

Given measurements:
- \( a_1 = 1.51 \)
- \( a_2 = 1.53 \)
- \( a_3 = 1.53 \)
- \( a_4 = 1.52 \)
- \( a_5 = 1.54 \)

Substituting the values:

\[ \text{Mean} = \frac{1.51 + 1.53 + 1.53 + 1.52 + 1.54}{5} = \frac{7.63}{5} = 1.526 \]

### Step 2: Calculate the Absolute Errors
Next, we calculate the absolute errors for each measurement using the formula:

\[ \Delta a_i = |a_i - \text{Mean}| \]

Calculating for each measurement:

- \( \Delta a_1 = |1.51 - 1.526| = 0.016 \)
- \( \Delta a_2 = |1.53 - 1.526| = 0.004 \)
- \( \Delta a_3 = |1.53 - 1.526| = 0.004 \)
- \( \Delta a_4 = |1.52 - 1.526| = 0.006 \)
- \( \Delta a_5 = |1.54 - 1.526| = 0.014 \)

### Step 3: Calculate the Average Absolute Error
The average absolute error is calculated by taking the mean of the absolute errors:

\[ \text{Average Absolute Error} = \frac{\Delta a_1 + \Delta a_2 + \Delta a_3 + \Delta a_4 + \Delta a_5}{5} \]

Substituting the values:

\[ \text{Average Absolute Error} = \frac{0.016 + 0.004 + 0.004 + 0.006 + 0.014}{5} = \frac{0.044}{5} = 0.0088 \]

### Step 4: Calculate the Relative Error
The relative error is calculated using the formula:

\[ \text{Relative Error} = \frac{\text{Average Absolute Error}}{\text{Mean}} \]

Substituting the values:

\[ \text{Relative Error} = \frac{0.0088}{1.526} \approx 0.00576 \]

### Step 5: Calculate the Percentage Error
The percentage error can be calculated by multiplying the relative error by 100:

\[ \text{Percentage Error} = \text{Relative Error} \times 100 \]

Substituting the values:

\[ \text{Percentage Error} = 0.00576 \times 100 \approx 0.576\% \]

### Final Results
- Mean: \( 1.526 \)
- Average Absolute Error: \( 0.0088 \)
- Relative Error: \( 0.00576 \)
- Percentage Error: \( 0.576\% \)