KAHIBARO
Table of Contents
Why uncertainty matters
Every physical measurement has some doubt in it. If you measure the length of a table, the temperature of water, or the time for a ball to fall, your result is never perfectly exact. The measured value is always an estimate of the true value.
Measurement uncertainty is the amount of doubt associated with a measurement. It tells us how much the measured value could reasonably differ from the true value. In physics, giving a measurement without its uncertainty is often incomplete, because the uncertainty tells us how reliable the number is.
A result such as $L = 2.4 \, \text{m}$ is less informative than
$$
L = 2.4 \pm 0.1 \, \text{m}
$$
This means the best estimate is $2.4 \, \text{m}$, and the actual value is expected to lie close to it, typically within about $0.1 \, \text{m}$.
A measurement should usually be reported as
$$
\text{measured value} \pm \text{uncertainty}
$$
with the correct unit.
Sources of uncertainty
Uncertainty comes from many practical limitations. Measuring instruments have limited precision. Human reaction time affects stopwatch readings. Environmental conditions such as temperature, vibration, or air flow can influence results. Even repeated measurements of the same quantity often give slightly different answers.
There are two broad kinds of measurement problems. One is random variation, where repeated readings scatter around a central value. The other is systematic shift, where all readings are pushed in the same wrong direction.
Random and systematic uncertainties
Random uncertainty causes measurements to fluctuate unpredictably from one trial to another. For example, timing a pendulum swing by hand may give values like $1.98 \, \text{s}$, $2.03 \, \text{s}$, and $2.01 \, \text{s}$. These differences are random.
Systematic uncertainty is different. It happens when there is a consistent bias. For example, if a ruler starts at $0.2 \, \text{cm}$ instead of zero, every length measured with it will be too small or too large by the same amount. If a stopwatch runs slow, all recorded times will be wrong in a similar way.
Random uncertainty can often be reduced by repeating measurements and averaging. Systematic uncertainty cannot be fixed by repetition alone. It must be found by checking the method, calibrating the instrument, or comparing with a trusted standard.
| Type of uncertainty | Main feature | Example | Can repeating help? |
|---|---|---|---|
| Random | Results scatter up and down | Human reaction time in timing | Yes, often |
| Systematic | Results are consistently shifted | Miscalibrated scale | No, not by itself |
Repeating a measurement reduces random uncertainty, but it does not remove systematic uncertainty.
Reading uncertainty from an instrument
A common way to estimate uncertainty is from the measuring instrument itself. If an instrument has markings, the smallest division gives a clue to the uncertainty.
For a ruler marked in millimeters, a reasonable uncertainty is often about half the smallest division:
$$
\Delta x \approx \pm 0.5 \, \text{mm}
$$
For digital instruments, the uncertainty is often taken as about one unit in the last displayed digit, unless the manufacturer gives a better value. If a digital balance reads $25.34 \, \text{g}$, one may estimate
$$
\Delta m \approx \pm 0.01 \, \text{g}
$$
This is a practical rule for beginners. In more advanced work, instrument manuals may provide a more detailed uncertainty.
Absolute uncertainty
Absolute uncertainty is the uncertainty written in the same unit as the measured quantity.
If a book’s width is measured as
$$
w = 18.2 \pm 0.1 \, \text{cm}
$$
then $0.1 \, \text{cm}$ is the absolute uncertainty.
Absolute uncertainty tells you the size of the possible error directly in physical units.
Relative and percentage uncertainty
Sometimes it is more useful to compare the uncertainty to the size of the measurement itself. This gives relative uncertainty:
$$
\text{relative uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}}
$$
For the book width above,
$$
\frac{0.1}{18.2} \approx 0.0055
$$
This can be written as a percentage uncertainty by multiplying by $100\%$:
$$
\text{percentage uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}} \times 100\%
$$
so
$$
\frac{0.1}{18.2} \times 100\% \approx 0.55\%
$$
Percentage uncertainty helps compare the quality of different measurements.
| Measurement | Absolute uncertainty | Relative uncertainty | Percentage uncertainty |
|---|---|---|---|
| $18.2 \, \text{cm}$ | $\pm 0.1 \, \text{cm}$ | $0.0055$ | $0.55\%$ |
| $2.0 \, \text{m}$ | $\pm 0.1 \, \text{m}$ | $0.05$ | $5\%$ |
Even though both measurements have the same absolute uncertainty of $0.1$ in their own units, the second is much less precise relative to its size.
$$
\text{percentage uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}} \times 100\%
$$
A small percentage uncertainty means a more precise measurement.
Repeated measurements and the mean
If you measure the same quantity many times and get slightly different results, a good estimate of the quantity is the mean, or average.
For measurements $x_1, x_2, x_3, \dots, x_n$, the mean is
$$
\bar{x} = \frac{x_1 + x_2 + x_3 + \cdots + x_n}{n}
$$
Suppose the measured times for an event are
$$
2.1 \, \text{s}, \quad 2.3 \, \text{s}, \quad 2.2 \, \text{s}, \quad 2.4 \, \text{s}
$$
Then the mean is
$$
\bar{t} = \frac{2.1 + 2.3 + 2.2 + 2.4}{4} = 2.25 \, \text{s}
$$
The spread of the results gives information about the uncertainty.
Estimating uncertainty from repeated values
For beginners, a simple estimate of uncertainty from repeated measurements is half the range:
$$
\Delta x \approx \frac{x_{\max} - x_{\min}}{2}
$$
In the previous timing example,
$$
x_{\max} = 2.4 \, \text{s}, \quad x_{\min} = 2.1 \, \text{s}
$$
so
$$
\Delta t \approx \frac{2.4 - 2.1}{2} = 0.15 \, \text{s}
$$
The result may be written as
$$
t = 2.25 \pm 0.15 \, \text{s}
$$
This method is simple and useful in introductory physics.
For repeated measurements, a common simple estimate is
$$
\Delta x \approx \frac{x_{\max} - x_{\min}}{2}
$$
and the final result is written using the mean value.
Reporting a measured result
A measured result should be written clearly, with value, uncertainty, and unit together. For example,
$$
V = 12.6 \pm 0.2 \, \text{V}
$$
This is much better than writing only $12.6$.
The uncertainty usually has one significant figure, or sometimes two if the first digit is $1$ or $2$. The measured value should then be rounded to the same decimal place as the uncertainty.
For example, this is good reporting:
$$
3.47 \pm 0.08 \, \text{m}
$$
and this is not well matched:
$$
3.47291 \pm 0.08 \, \text{m}
$$
because the value is written with more precision than the uncertainty supports.
Uncertainty in comparisons
Uncertainty helps us decide whether two measurements agree. Suppose one group measures a rod length as
$$
5.2 \pm 0.1 \, \text{cm}
$$
and another group finds
$$
5.3 \pm 0.1 \, \text{cm}
$$
These results are close enough that they may agree within uncertainty. But if one result is
$$
5.2 \pm 0.1 \, \text{cm}
$$
and another is
$$
5.9 \pm 0.1 \, \text{cm}
$$
then they likely disagree, and something should be checked.
Uncertainty therefore helps judge whether differences are meaningful or just due to measurement variation.
Reducing uncertainty
In practical work, uncertainty can often be reduced by improving technique. A more precise instrument helps. Taking repeated measurements helps reduce random effects. Measuring a larger total quantity and dividing can reduce relative uncertainty, such as timing ten oscillations instead of one. Careful calibration and correct zeroing help reduce systematic errors.
Final idea
Measurement uncertainty is not a mistake to hide. It is a necessary part of doing physics honestly. It tells us the quality of data, the limits of our instruments, and how much confidence we should place in a result.
In physics, a number from measurement is incomplete unless its uncertainty is known or understood.