Step 1: Understanding the Concept:
When an experiment yields multiple readings for a single physical quantity, the most probable value is represented by the arithmetic mean.
To quantify the uncertainty or precision of the measurements, we calculate the mean absolute error.
The final result is standardized in the format: \(\text{Mean Value} \pm \text{Mean Absolute Error}\).
Step 2: Key Formula or Approach:
1. The arithmetic mean (\(t_m\)) is calculated as:
\[ t_m = \frac{t_1 + t_2 + \dots + t_n}{n} \]
2. The absolute error for each reading (\(\Delta t_i\)) is the absolute difference from the mean:
\[ \Delta t_i = |t_m - t_i| \]
3. The mean absolute error (\(\Delta t_m\)) is the average of these absolute errors:
\[ \Delta t_m = \frac{\Delta t_1 + \Delta t_2 + \dots + \Delta t_n}{n} \]
Step 3: Detailed Explanation:
The recorded time measurements are: \(t_1 = 30\text{ s}, t_2 = 32\text{ s}, t_3 = 35\text{ s}, t_4 = 35\text{ s}\).
First, we find the mean time (\(t_m\)):
\[ t_m = \frac{30 + 32 + 35 + 35}{4} = \frac{132}{4} = 33\text{ s} \]
Next, we evaluate the absolute error for each individual measurement:
\[ \Delta t_1 = |33 - 30| = 3\text{ s} \]
\[ \Delta t_2 = |33 - 32| = 1\text{ s} \]
\[ \Delta t_3 = |33 - 35| = 2\text{ s} \]
\[ \Delta t_4 = |33 - 35| = 2\text{ s} \]
Now, calculate the mean absolute error (\(\Delta t_m\)):
\[ \Delta t_m = \frac{3 + 1 + 2 + 2}{4} = \frac{8}{4} = 2\text{ s} \]
The minimum division (least count) of the clock is \(1\text{ s}\). Since our calculated mean absolute error (\(2\text{ s}\)) is larger than the least count, the error in measurement dominates the precision limit.
The final correct mean time is expressed by combining the mean value and the mean absolute error.
Final expressed value = \(t_m \pm \Delta t_m = (33 \pm 2)\text{ s}\).
Step 4: Final Answer:
The correct mean time is \((33 \pm 2)\).