The values for the diameter of a wire as measured by a screw gauge were found to be 0.026 cm, 0.028 cm, 0.029 cm, 0.027 cm, 0.024 cm and 0.027 cm. Find the mean value and the relative error.

Answer

To solve the problem of finding the mean value and the relative error for the diameter of a wire measured by a screw gauge, we will follow these steps:

### Step 1: List the Measurements
The measurements given for the diameter of the wire are:
- 0.026 cm
- 0.028 cm
- 0.029 cm
- 0.027 cm
- 0.024 cm
- 0.027 cm

### Step 2: Calculate the Mean Value
To find the mean value, we sum all the measurements and divide by the number of measurements.

1. **Sum of Measurements**:
\[ \text{Sum} = 0.026 + 0.028 + 0.029 + 0.027 + 0.024 + 0.027 \]
\[ \text{Sum} = 0.026 + 0.028 + 0.029 + 0.027 + 0.024 + 0.027 = 0.161 \text{ cm} \]

2. **Number of Measurements**:
There are 6 measurements.

3. **Mean Value Calculation**:
\[ \text{Mean} = \frac{\text{Sum}}{\text{Number of Measurements}} = \frac{0.161}{6} \approx 0.0268333 \text{ cm} \]
Rounding this to three decimal places gives:
\[ \text{Mean} \approx 0.027 \text{ cm} \]

### Step 3: Calculate the Absolute Error for Each Measurement
The absolute error for each measurement is calculated by taking the difference between each measurement and the mean value.

1. **Calculating Absolute Errors**:
- For 0.026 cm:
\[ |0.027 - 0.026| = 0.001 \text{ cm} \]
- For 0.028 cm:
\[ |0.027 - 0.028| = 0.001 \text{ cm} \]
- For 0.029 cm:
\[ |0.027 - 0.029| = 0.002 \text{ cm} \]
- For 0.027 cm:
\[ |0.027 - 0.027| = 0.000 \text{ cm} \]
- For 0.024 cm:
\[ |0.027 - 0.024| = 0.003 \text{ cm} \]
- For 0.027 cm:
\[ |0.027 - 0.027| = 0.000 \text{ cm} \]

### Step 4: Calculate the Mean Absolute Error
Now, we find the mean of the absolute errors calculated above.

1. **Sum of Absolute Errors**:
\[ \text{Sum of Absolute Errors} = 0.001 + 0.001 + 0.002 + 0.000 + 0.003 + 0.000 = 0.007 \text{ cm} \]

2. **Mean Absolute Error Calculation**:
\[ \text{Mean Absolute Error} = \frac{\text{Sum of Absolute Errors}}{\text{Number of Measurements}} = \frac{0.007}{6} \approx 0.0011667 \text{ cm} \]

### Step 5: Calculate the Relative Error
The relative error is calculated using the formula:
\[ \text{Relative Error} = \frac{\text{Mean Absolute Error}}{\text{Mean Value}} \]

1. **Relative Error Calculation**:
\[ \text{Relative Error} = \frac{0.0011667}{0.027} \approx 0.0433 \]

### Summary of Results
- **Mean Value**: 0.027 cm
- **Relative Error**: 0.0433 (or 4.33%)