In an experiment the refractive index of glass was observed to be `1.45 , 1.56 , 1.54 , 1.44 , 1.54 , and 1.53`. Calculate
(a). Mean value of refractive index
(b). Mean absolute error
( c ) Fractional error
(d) Percentage error
(e) Express the result in terms of absolute error and percentage error

Explanation

To solve the problem step by step, we will calculate the mean value of the refractive index, the mean absolute error, the fractional error, the percentage error, and finally express the result in terms of absolute and percentage error.

### Step 1: Calculate the Mean Value of Refractive Index
To find the mean value, we will sum all the observed refractive indices and divide by the total number of observations.

Given values:
- 1.45, 1.56, 1.54, 1.44, 1.54, 1.53

**Calculation:**
\[ \text{Mean} = \frac{1.45 + 1.56 + 1.54 + 1.44 + 1.54 + 1.53}{6} \]
\[ = \frac{9.12}{6} = 1.52 \]

### Step 2: Calculate the Mean Absolute Error
The mean absolute error is calculated by finding the absolute error for each observation and then taking the mean of those errors.

**Errors Calculation:**
1. For 1.45: \( |1.52 - 1.45| = 0.07 \)
2. For 1.56: \( |1.52 - 1.56| = 0.04 \)
3. For 1.54: \( |1.52 - 1.54| = 0.02 \)
4. For 1.44: \( |1.52 - 1.44| = 0.08 \)
5. For 1.54: \( |1.52 - 1.54| = 0.02 \)
6. For 1.53: \( |1.52 - 1.53| = 0.01 \)

**Sum of Absolute Errors:**
\[ \text{Total Error} = 0.07 + 0.04 + 0.02 + 0.08 + 0.02 + 0.01 = 0.24 \]

**Mean Absolute Error:**
\[ \text{Mean Absolute Error} = \frac{0.24}{6} = 0.04 \]

### Step 3: Calculate the Fractional Error
The fractional error is calculated by dividing the mean absolute error by the mean value.

**Calculation:**
\[ \text{Fractional Error} = \frac{\text{Mean Absolute Error}}{\text{Mean Value}} = \frac{0.04}{1.52} \approx 0.0263 \]

### Step 4: Calculate the Percentage Error
The percentage error is calculated by multiplying the fractional error by 100.

**Calculation:**
\[ \text{Percentage Error} = \text{Fractional Error} \times 100 \approx 0.0263 \times 100 \approx 2.63\% \]

### Step 5: Express the Result in Terms of Absolute and Percentage Error
We can express the result as follows:

**Final Result:**
\[ \text{Refractive Index} \, (\mu) = 1.52 \pm 0.04 \quad \text{or} \quad 1.52 \pm 2.63\% \]

### Summary of Results:
- (a) Mean Value of Refractive Index: **1.52**
- (b) Mean Absolute Error: **0.04**
- (c) Fractional Error: **0.0263**
- (d) Percentage Error: **2.63%**
- (e) Result: **1.52 ± 0.04** or **1.52 ± 2.63%**