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 of the diameter of a wire measured by a screw gauge, we will follow these steps:

### Step 1: List the Measurements
The given measurements of 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 (\( x_{\text{mean}} \)), we sum all the measurements and divide by the number of measurements.

\[ x_{\text{mean}} = \frac{0.026 + 0.028 + 0.029 + 0.027 + 0.024 + 0.027}{6} \]

Calculating the sum:
\[ 0.026 + 0.028 + 0.029 + 0.027 + 0.024 + 0.027 = 0.161 \text{ cm} \]

Now, dividing by the number of measurements (6):
\[ x_{\text{mean}} = \frac{0.161}{6} \approx 0.0268333 \text{ cm} \]

Rounding off, we get:
\[ x_{\text{mean}} \approx 0.027 \text{ cm} \]

### Step 3: Calculate the Absolute Errors
Next, we calculate the absolute error for each measurement, which is the absolute difference between each measurement and the mean value.

- For \( x_1 = 0.026 \):
\[ |0.026 - 0.027| = 0.001 \text{ cm} \]

- For \( x_2 = 0.028 \):
\[ |0.028 - 0.027| = 0.001 \text{ cm} \]

- For \( x_3 = 0.029 \):
\[ |0.029 - 0.027| = 0.002 \text{ cm} \]

- For \( x_4 = 0.027 \):
\[ |0.027 - 0.027| = 0.000 \text{ cm} \]

- For \( x_5 = 0.024 \):
\[ |0.024 - 0.027| = 0.003 \text{ cm} \]

- For \( x_6 = 0.027 \):
\[ |0.027 - 0.027| = 0.000 \text{ cm} \]

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

Sum of absolute errors:
\[ 0.001 + 0.001 + 0.002 + 0.000 + 0.003 + 0.000 = 0.007 \text{ cm} \]

Mean absolute error:
\[ \text{Mean Absolute Error} = \frac{0.007}{6} \approx 0.0011667 \text{ cm} \]

Rounding off, we get:
\[ \text{Mean Absolute Error} \approx 0.001 \text{ cm} \]

### Step 5: Calculate the Relative Error
Finally, we calculate the relative error, which is the mean absolute error divided by the mean value.

\[ \text{Relative Error} = \frac{\text{Mean Absolute Error}}{x_{\text{mean}}} = \frac{0.001}{0.027} \]

Calculating this gives:
\[ \text{Relative Error} \approx 0.037037 \text{ (dimensionless)} \]

### Final Results
- Mean Value: \( 0.027 \text{ cm} \)
- Relative Error: \( 0.037 \)