The diameter of a wire as measured by a screw gauge was found to be 0.026 cm, 0.028 cm, 0.029 cm, 0.027cm, 0.024cm and 0.027 cm. Calculate
(i) mean value of diameter
(ii) mean absoulte error
(iii) relative error (iv) percentage error. Also 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 diameter, mean absolute error, relative error, and percentage error based on the measurements provided.

### Given Measurements:
- \( d_1 = 0.026 \, \text{cm} \)
- \( d_2 = 0.028 \, \text{cm} \)
- \( d_3 = 0.029 \, \text{cm} \)
- \( d_4 = 0.027 \, \text{cm} \)
- \( d_5 = 0.024 \, \text{cm} \)
- \( d_6 = 0.027 \, \text{cm} \)

### Step 1: Calculate the Mean Value of Diameter
The mean value \( \bar{d} \) is calculated using the formula:

\[ \bar{d} = \frac{d_1 + d_2 + d_3 + d_4 + d_5 + d_6}{6} \]

Calculating the sum of the measurements:

\[ d_1 + d_2 + d_3 + d_4 + d_5 + d_6 = 0.026 + 0.028 + 0.029 + 0.027 + 0.024 + 0.027 = 0.161 \, \text{cm} \]

Now, dividing by 6:

\[ \bar{d} = \frac{0.161}{6} = 0.0268333 \, \text{cm} \approx 0.027 \, \text{cm} \quad (\text{rounded to 3 decimal places}) \]

### Step 2: Calculate the Mean Absolute Error
The mean absolute error (MAE) is calculated by finding the absolute deviations from the mean and then averaging those deviations.

1. Calculate the absolute errors:
- \( \Delta d_1 = |d_1 - \bar{d}| = |0.026 - 0.027| = 0.001 \, \text{cm} \)
- \( \Delta d_2 = |d_2 - \bar{d}| = |0.028 - 0.027| = 0.001 \, \text{cm} \)
- \( \Delta d_3 = |d_3 - \bar{d}| = |0.029 - 0.027| = 0.002 \, \text{cm} \)
- \( \Delta d_4 = |d_4 - \bar{d}| = |0.027 - 0.027| = 0.000 \, \text{cm} \)
- \( \Delta d_5 = |d_5 - \bar{d}| = |0.024 - 0.027| = 0.003 \, \text{cm} \)
- \( \Delta d_6 = |d_6 - \bar{d}| = |0.027 - 0.027| = 0.000 \, \text{cm} \)

2. Calculate the mean absolute error:

\[ \text{MAE} = \frac{\Delta d_1 + \Delta d_2 + \Delta d_3 + \Delta d_4 + \Delta d_5 + \Delta d_6}{6} \]

Calculating the sum of absolute errors:

\[ \Delta d_1 + \Delta d_2 + \Delta d_3 + \Delta d_4 + \Delta d_5 + \Delta d_6 = 0.001 + 0.001 + 0.002 + 0.000 + 0.003 + 0.000 = 0.007 \, \text{cm} \]

Now, dividing by 6:

\[ \text{MAE} = \frac{0.007}{6} \approx 0.00116667 \, \text{cm} \approx 0.001 \, \text{cm} \quad (\text{rounded}) \]

### Step 3: Calculate the Relative Error
The relative error is calculated using the formula:

\[ \text{Relative Error} = \frac{\text{MAE}}{\bar{d}} \]

Substituting the values:

\[ \text{Relative Error} = \frac{0.001}{0.027} \approx 0.037037 \quad (\text{approximately}) \]

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

\[ \text{Percentage Error} = \text{Relative Error} \times 100 \approx 0.037037 \times 100 \approx 3.7037\% \approx 3.7\% \]

### Final Results
1. Mean Value of Diameter: \( 0.027 \, \text{cm} \)
2. Mean Absolute Error: \( 0.001 \, \text{cm} \)
3. Relative Error: \( 0.037 \)
4. Percentage Error: \( 3.7\% \)

### Expressing Results in Terms of Absolute and Percentage Error
- In terms of absolute error: \( \bar{d} \pm \text{MAE} = 0.027 \pm 0.001 \, \text{cm} \)
- In terms of percentage error: \( \bar{d} \pm \text{Percentage Error} = 0.027 \pm 3.7\% \)