Time intervals measured by clock give following readings: 1.25s, 1.24s, 1.27s, 1.21s and 1.28s. What is percentage relative error of observations?

Options

A. 0.02
B. 0.04
C. 0.16
D. 0.016

Answer

To find the percentage relative error of the observations, we can follow these steps:

### Step 1: Calculate the Mean Value of Observations
We have the following time interval readings:
- 1.25 s
- 1.24 s
- 1.27 s
- 1.21 s
- 1.28 s

To find the mean value, we sum all the readings and divide by the number of readings.

\[ \text{Mean} = \frac{1.25 + 1.24 + 1.27 + 1.21 + 1.28}{5} \]

Calculating the sum:

\[ 1.25 + 1.24 + 1.27 + 1.21 + 1.28 = 6.25 \]

Now, dividing by the number of readings (5):

\[ \text{Mean} = \frac{6.25}{5} = 1.25 \text{ s} \]

### Step 2: Calculate the Absolute Errors
Next, we calculate the absolute error for each observation by subtracting the mean from each reading:

- For 1.25 s:
\[ e_1 = 1.25 - 1.25 = 0 \]

- For 1.24 s:
\[ e_2 = 1.24 - 1.25 = -0.01 \]

- For 1.27 s:
\[ e_3 = 1.27 - 1.25 = 0.02 \]

- For 1.21 s:
\[ e_4 = 1.21 - 1.25 = -0.04 \]

- For 1.28 s:
\[ e_5 = 1.28 - 1.25 = 0.03 \]

### Step 3: Calculate the Sum of Absolute Errors
Now, we take the absolute values of the errors and sum them up:

\[ \text{Sum of Absolute Errors} = |e_1| + |e_2| + |e_3| + |e_4| + |e_5| \]

Calculating the absolute values:

\[ |e_1| = 0, \quad |e_2| = 0.01, \quad |e_3| = 0.02, \quad |e_4| = 0.04, \quad |e_5| = 0.03 \]

Now summing them:

\[ \text{Sum of Absolute Errors} = 0 + 0.01 + 0.02 + 0.04 + 0.03 = 0.10 \]

### Step 4: Calculate the Relative Error
The relative error is calculated using the formula:

\[ \text{Relative Error} = \frac{\text{Sum of Absolute Errors}}{\text{Mean}} \times \frac{100}{n} \]

Where \( n \) is the number of observations (5 in this case).

Substituting the values:

\[ \text{Relative Error} = \frac{0.10}{1.25} \times \frac{100}{5} \]

Calculating:

\[ \text{Relative Error} = \frac{0.10}{1.25} \times 20 = 0.08 \times 20 = 1.6\% \]

### Final Answer
The percentage relative error of the observations is **1.6%**.

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