Animated Solution for Physics - Physics and Measurement: A student measures the time period of 100 oscillations of a simple pendulum four times. The data set is $90\text{s}$, $91\text{s}$, $92\text{s}$ and $95\text{s}$. If the minimum division in the measuring clock is $1\text{s}$, then the reported mean time should be

Imagine you are in a physics lab, carefully timing the swings of a simple pendulum. You record the time for 100 oscillations four separate times, and your stopwatch gives you the following data set: $90\text{s}$, $91\text{s}$, $92\text{s}$, and $95\text{s}$.

You also note a crucial detail about your instrument: the minimum division on your measuring clock is $1\text{s}$. This is known as the least count, and it represents the absolute limit of your clock's precision. Our mission is to take these raw numbers and present a scientifically accurate reported mean time.

Finding the Mean

First things first, we need to find the best estimate for the true time period. In experimental physics, the arithmetic mean of our readings serves as this best estimate.

We calculate the mean, denoted as $\bar{x}$, by summing all our individual readings and dividing by the total number of readings ($N$):

Substituting our values:

So, our best estimate for the time period is exactly $92\text{s}$.

Mean Absolute Deviation

Now, we must determine how reliable our mean is. How far off, on average, were our individual readings from this mean? To find out, we calculate the absolute deviation for each reading. We take the absolute value because we care about the magnitude of the error, regardless of whether the reading was too high or too low.

Next, we find the average of these deviations, known as the mean absolute deviation ($\Delta \bar{x}$):

Our calculations tell us that, on average, our measurements deviate by $1.5\text{s}$.

Here is where many students make a silly mistake. You might be tempted to report the final answer as $92\pm 1.5\text{s}$. But wait! Remember our clock's least count? It is $1\text{s}$.

It is a fundamental rule of error analysis that you cannot report an error with more precision than your measuring instrument allows. Furthermore, errors are typically rounded to one significant figure because an error is, by definition, an estimate of uncertainty.

Therefore, we must round our calculated error of $1.5\text{s}$ to the nearest integer, which gives us $2\text{s}$.

Finally, we combine our arithmetic mean and our correctly rounded error to state our final reported value:

This elegant notation tells the world that our best estimate is $92\text{s}$, and we are confident the true value lies somewhere between $90\text{s}$ and $94\text{s}$.