A student makes measurements of the diameter of a wire with the help a screw gauge and he gets the following readings : `0.38, 0.40, 0.39, 0.37, 0.41, 0.40, 0.38, 0.39, 0.40` and `0.41` mm. Calculate the average error and standard deviation of his measurements.

Answer

To solve the problem of calculating the average error and standard deviation of the measurements of the diameter of a wire, we will follow these steps:

### Step 1: List the Measurements
The measurements given are:
- 0.38 mm
- 0.40 mm
- 0.39 mm
- 0.37 mm
- 0.41 mm
- 0.40 mm
- 0.38 mm
- 0.39 mm
- 0.40 mm
- 0.41 mm

### Step 2: Calculate the Average (Mean) Measurement
To calculate the average (mean) measurement, we use the formula:
\[ \bar{d} = \frac{d_1 + d_2 + d_3 + ... + d_n}{n} \]
Where \( n \) is the number of measurements.

Calculating the sum:
\[ 0.38 + 0.40 + 0.39 + 0.37 + 0.41 + 0.40 + 0.38 + 0.39 + 0.40 + 0.41 = 3.90 \text{ mm} \]
Now, divide by the number of measurements (10):
\[ \bar{d} = \frac{3.90}{10} = 0.390 \text{ mm} \]

### Step 3: Calculate the Absolute Errors
Next, we calculate the absolute error for each measurement:
\[ \text{Absolute Error} = |d_i - \bar{d}| \]
Calculating for each measurement:
- For 0.38 mm: \( |0.38 - 0.390| = 0.010 \)
- For 0.40 mm: \( |0.40 - 0.390| = 0.010 \)
- For 0.39 mm: \( |0.39 - 0.390| = 0.000 \)
- For 0.37 mm: \( |0.37 - 0.390| = 0.020 \)
- For 0.41 mm: \( |0.41 - 0.390| = 0.020 \)
- For 0.40 mm: \( |0.40 - 0.390| = 0.010 \)
- For 0.38 mm: \( |0.38 - 0.390| = 0.010 \)
- For 0.39 mm: \( |0.39 - 0.390| = 0.000 \)
- For 0.40 mm: \( |0.40 - 0.390| = 0.010 \)
- For 0.41 mm: \( |0.41 - 0.390| = 0.020 \)

### Step 4: Calculate the Average Error
The average error is calculated as:
\[ \text{Average Error} = \frac{\sum |d_i - \bar{d}|}{n} \]
Calculating the sum of absolute errors:
\[ 0.010 + 0.010 + 0.000 + 0.020 + 0.020 + 0.010 + 0.010 + 0.000 + 0.010 + 0.020 = 0.110 \]
Now, divide by the number of measurements (10):
\[ \text{Average Error} = \frac{0.110}{10} = 0.011 \text{ mm} \]

### Step 5: Calculate the Standard Deviation
The standard deviation is calculated using the formula:
\[ \sigma = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}} \]
Calculating each squared error:
- For 0.38 mm: \( (0.38 - 0.390)^2 = 0.0001 \)
- For 0.40 mm: \( (0.40 - 0.390)^2 = 0.0001 \)
- For 0.39 mm: \( (0.39 - 0.390)^2 = 0.0000 \)
- For 0.37 mm: \( (0.37 - 0.390)^2 = 0.0004 \)
- For 0.41 mm: \( (0.41 - 0.390)^2 = 0.0004 \)
- For 0.40 mm: \( (0.40 - 0.390)^2 = 0.0001 \)
- For 0.38 mm: \( (0.38 - 0.390)^2 = 0.0001 \)
- For 0.39 mm: \( (0.39 - 0.390)^2 = 0.0000 \)
- For 0.40 mm: \( (0.40 - 0.390)^2 = 0.0001 \)
- For 0.41 mm: \( (0.41 - 0.390)^2 = 0.0004 \)

Calculating the sum of squared errors:
\[ 0.0001 + 0.0001 + 0.0000 + 0.0004 + 0.0004 + 0.0001 + 0.0001 + 0.0000 + 0.0001 + 0.0004 = 0.0017 \]
Now, divide by \( n - 1 = 9 \):
\[ \sigma = \sqrt{\frac{0.0017}{9}} = \sqrt{0.00018889} \approx 0.0137 \text{ mm} \]

### Final Results
- **Average Error**: \( 0.011 \text{ mm} \)
- **Standard Deviation**: \( 0.0137 \text{ mm} \)

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