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 given, we will follow these steps:

### Step 1: Calculate the Mean Time (t_mean)
We have the readings:
- t1 = 1.25 s
- t2 = 1.24 s
- t3 = 1.27 s
- t4 = 1.21 s
- t5 = 1.28 s

The mean time (t_mean) is calculated using the formula:
\[ t_{\text{mean}} = \frac{t_1 + t_2 + t_3 + t_4 + t_5}{5} \]

Substituting the values:
\[ t_{\text{mean}} = \frac{1.25 + 1.24 + 1.27 + 1.21 + 1.28}{5} = \frac{6.25}{5} = 1.25 \text{ s} \]

### Step 2: Calculate the Absolute Deviations (Δt_i)
Next, we calculate the absolute deviations (Δt_i) from the mean for each reading:
- Δt1 = |t_mean - t1| = |1.25 - 1.25| = 0.00 s
- Δt2 = |t_mean - t2| = |1.25 - 1.24| = 0.01 s
- Δt3 = |t_mean - t3| = |1.25 - 1.27| = 0.02 s
- Δt4 = |t_mean - t4| = |1.25 - 1.21| = 0.04 s
- Δt5 = |t_mean - t5| = |1.25 - 1.28| = 0.03 s

### Step 3: Calculate the Mean Absolute Deviation (Δt_mean)
Now, we calculate the mean of these absolute deviations:
\[ \Delta t_{\text{mean}} = \frac{\Delta t_1 + \Delta t_2 + \Delta t_3 + \Delta t_4 + \Delta t_5}{5} \]

Substituting the values:
\[ \Delta t_{\text{mean}} = \frac{0.00 + 0.01 + 0.02 + 0.04 + 0.03}{5} = \frac{0.10}{5} = 0.02 \text{ s} \]

### Step 4: Calculate the Percentage Relative Error
Finally, we can calculate the percentage relative error using the formula:
\[ \text{Percentage Relative Error} = \left( \frac{\Delta t_{\text{mean}}}{t_{\text{mean}}} \right) \times 100\% \]

Substituting the values:
\[ \text{Percentage Relative Error} = \left( \frac{0.02}{1.25} \right) \times 100\% = 1.6\% \]

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

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