Measurements in Physics

When measuring any physical quantity, you never obtain the exact “true value” in absolute terms but rather an estimate with a certain degree of uncertainty.

For this reason, every measurement is expressed as a range, showing both the measurement result and its associated uncertainty.

$$ x = x \pm e_x $$

This format helps convey that, with a reasonable level of confidence, the true value of \( x \) lies somewhere between \( x - e_x \) and \( x + e_x \).

How do you determine the best estimate and uncertainty? The approach depends on whether you have a single measurement or multiple repeated measurements of the same quantity.

Single Measurement Scenario

With only one measurement, determining the best estimate and uncertainty is straightforward:

• The best estimate of the quantity is simply the measured value itself.
• The absolute error is defined by the sensitivity of the measuring instrument, meaning the smallest change it can detect.

Here’s a practical example to illustrate.

Imagine you’re measuring someone’s weight.

In this case, the instrument is a scale, with a sensitivity of 0.1 kg (the smallest detectable difference) and a maximum capacity of 200 kg.

If the scale reads 74.0 kg, then:

• The best estimate for the weight is 74.0 kg;
• The absolute error, based on the instrument’s sensitivity, is \( e_x = 0.1 \) kg.

So, the measurement result is expressed as:

$$ x = (74.0 \pm 0.1) \text{ kg} $$

This indicates that the actual weight is likely somewhere between:

$$ (74.0 - 0.1) \ \text{ kg} \lt x \lt (74.0 + 0.1) \ \text{ kg} $$

$$ 73.9 \ \text{ kg} \lt x \lt 74.1 \ \text{ kg} $$

In essence, measuring a physical quantity provides a value that aims to best represent reality, paired with an indication of its accuracy.

The uncertainty in a measurement is crucial for assessing its reliability: the smaller the error, the greater the precision. While simple, the act of measurement brings us closer to reality, with the understanding that there’s always some degree of uncertainty, whether in physics or in life.

Multiple Measurements Scenario

When you measure the same quantity multiple times, it’s normal to get slightly different results due to minor accidental variations or fluctuations.

In this case, the best estimate of the quantity is the average of the measured values.

Let’s say you’ve taken \( n \) measurements, resulting in values \( x_1, x_2, \ldots, x_n \); the average, denoted as \( \bar{x} \), is calculated as:

$$ \bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n} $$

The absolute error here is no longer just the instrument’s sensitivity but also depends on the spread of the measured values.

To estimate the error for a series of repeated measurements, we use the standard deviation of the mean, which indicates how the values vary around the average and is calculated as:

$$ s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} $$

Where \( x_i \) represents each measurement, and \( \bar{x} \) is the average.

We can also calculate the standard error of the mean (\( e_x \)) as:

$$ e_x = \frac{s}{\sqrt{n}} $$

This value represents the uncertainty in the average of the measurements, which decreases as the number of measurements \( n \) increases.

Practical Example

Suppose you measure the length of an object five times, getting these results:

\( 10.2 \) cm, \( 10.3 \) cm, \( 10.2 \) cm, \( 10.4 \) cm, and \( 10.1 \) cm.

Here’s the process: first, calculate the average:

$$ \bar{x} = \frac{10.2 + 10.3 + 10.2 + 10.4 + 10.1}{5} = 10.24 \, \text{cm} $$

Then calculate the standard deviation (\( s \)):

$$ s = \sqrt{\frac{(10.2 - 10.24)^2 + (10.3 - 10.24)^2 + (10.2 - 10.24)^2 + (10.4 - 10.24)^2 + (10.1 - 10.24)^2}{5 - 1}} $$

After completing the calculation, you’ll find that \( s \) is approximately \( 0.1 \) cm.

$$ s = 0.1 \ cm $$

In this case, the standard error of the mean is:

$$ e_x = \frac{s}{\sqrt{5}} \approx \frac{0.1}{\sqrt{5}} \approx 0.045 \, \text{cm} $$

So the final measurement, considering the error, is:

$$ x = (10.24 \pm 0.045) \, \text{cm} $$

This means the object’s true length is likely between \( 10.24 - 0.045 = 10.195 \) cm and \( 10.24 + 0.045 = 10.285 \) cm.

The average value of the measurements provides the best estimate, while the standard deviation of the mean indicates the uncertainty associated with the result.

Note that with repeated measurements of the same physical quantity, calculating the best estimate and associated uncertainty offers a more accurate approximation of the true value. This is because taking repeated measurements helps reduce the uncertainty due to random errors.

In conclusion, taking multiple measurements is more involved but yields a more reliable result.

This approach is especially helpful when higher accuracy is desired.