In an experiment the valno of refractive index of glass was found to be 1.54, 1.53, 1.44,1.54, 1.56 and 1.45 m in successive measurement. Calculate (i) the mean vlaue of refractive index (ii) absolute error of each measurement (iii) mean absolute error (iv) relative error and (v) percentage error.

Explanation

To solve the problem step by step, we will follow the instructions given in the question and calculate each required value systematically.

### Step 1: Calculate the Mean Value of the Refractive Index

The refractive index values obtained from the experiment are:
- A1 = 1.54
- A2 = 1.53
- A3 = 1.44
- A4 = 1.54
- A5 = 1.56
- A6 = 1.45

To find the mean value (A_mean), we use the formula:
\[ A_{\text{mean}} = \frac{A_1 + A_2 + A_3 + A_4 + A_5 + A_6}{6} \]

Substituting the values:
\[ A_{\text{mean}} = \frac{1.54 + 1.53 + 1.44 + 1.54 + 1.56 + 1.45}{6} = \frac{9.06}{6} = 1.51 \]

### Step 2: Calculate the Absolute Error of Each Measurement

The absolute error (ΔA) for each measurement is calculated as:
\[ \Delta A_i = A_{\text{mean}} - A_i \]
Calculating for each measurement:
- For A1: \(\Delta A_1 = 1.51 - 1.54 = -0.03\)
- For A2: \(\Delta A_2 = 1.51 - 1.53 = -0.02\)
- For A3: \(\Delta A_3 = 1.51 - 1.44 = 0.07\)
- For A4: \(\Delta A_4 = 1.51 - 1.54 = -0.03\)
- For A5: \(\Delta A_5 = 1.51 - 1.56 = -0.05\)
- For A6: \(\Delta A_6 = 1.51 - 1.45 = 0.06\)

### Step 3: Calculate the Mean Absolute Error

The mean absolute error (ΔA_mean) is calculated as:
\[ \Delta A_{\text{mean}} = \frac{|\Delta A_1| + |\Delta A_2| + |\Delta A_3| + |\Delta A_4| + |\Delta A_5| + |\Delta A_6|}{6} \]

Calculating the absolute values:
\[ \Delta A_{\text{mean}} = \frac{0.03 + 0.02 + 0.07 + 0.03 + 0.05 + 0.06}{6} = \frac{0.26}{6} \approx 0.0433 \approx 0.04 \text{ (rounded)} \]

### Step 4: Calculate the Relative Error

The relative error is given by:
\[ \text{Relative Error} = \frac{\Delta A_{\text{mean}}}{A_{\text{mean}}} \]
Substituting the values:
\[ \text{Relative Error} = \frac{0.04}{1.51} \approx 0.0265 \approx 0.03 \text{ (rounded)} \]

### Step 5: Calculate the Percentage Error

The percentage error is calculated as:
\[ \text{Percentage Error} = \left(\frac{\Delta A_{\text{mean}}}{A_{\text{mean}}}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Error} = \left(\frac{0.04}{1.51}\right) \times 100 \approx 2.65\% \approx 3\% \text{ (rounded)} \]

### Summary of Results
1. Mean value of refractive index: **1.51**
2. Absolute errors: **-0.03, -0.02, 0.07, -0.03, -0.05, 0.06**
3. Mean absolute error: **0.04**
4. Relative error: **0.03**
5. Percentage error: **3%**