Uncertainty in Repeated Measurements: Measurement in Scientific Work - Physics (Grade 10)

Different Data Does Not Mean Failure

When one object is measured several times, the results can differ slightly. That is normal. A hand can press the tool with a different force, an eye can read the scale from a slightly different angle, and the object can shift a little.

Measurement uncertainty is an estimate of how far a measurement result may reasonably be from the value we report. For repeated measurements, uncertainty mainly reads the spread of the data.

Repeated measurements help reveal random error, meaning small changes that move up and down from one reading to another. Systematic error, such as a tool that does not start at zero or a damaged scale, still needs tool checking or calibration.

The Mean Represents the Data

Suppose $5$ students measure the diameter of a bottle cap with a vernier caliper. Every reading uses the same unit, $\text{cm}$ .

Data	Diameter
$d_1$	$3.12 \text{ cm}$
$d_2$	$3.14 \text{ cm}$
$d_3$	$3.15 \text{ cm}$
$d_4$	$3.11 \text{ cm}$
$d_5$	$3.14 \text{ cm}$

The most reasonable single value for repeated data is the mean.

\bar d=\frac{\sum_{i=1}^{N} d_i}{N}

For the data above:

\begin{aligned} \bar d &= \frac{3.12+3.14+3.15+3.11+3.14}{5}\text{ cm} \\ &= 3.132 \text{ cm} \end{aligned}

The mean is not enough. We also need to know whether the data cluster tightly around the mean or spread farther away.

The Spread Around the Mean

For repeated data, sample spread can be calculated from the distance between each reading and the mean.

s_d=\sqrt{\frac{\sum_{i=1}^{N}(d_i-\bar d)^2}{N-1}}

The value $N-1$ is used because the measurement data is a small sample, not every possible measurement that could happen.

\begin{aligned} s_d &= \sqrt{\frac{(-0.012)^2+(0.008)^2+(0.018)^2+(-0.022)^2+(0.008)^2}{4}}\text{ cm} \\ &= 0.0164\text{ cm} \end{aligned}

If we report the diameter with the spread of the repeated data, the result becomes:

d=(3.13 \pm 0.02)\text{ cm}

The value $3.13 \text{ cm}$ represents the data. The value $0.02 \text{ cm}$ reminds the reader that the repeated readings were not exactly the same.

The Compact Formula That Gets Confused

School books sometimes write repeated-data spread in the compact form below.

\Delta x=\frac{1}{\sqrt{N}}\sqrt{\frac{N\sum_{i=1}^{N}x_i^2-\left(\sum_{i=1}^{N}x_i\right)^2}{N-1}}

That formula is equivalent to the sample standard deviation:

\Delta x=s_x=\sqrt{\frac{\sum_{i=1}^{N}(x_i-\bar x)^2}{N-1}}

So the $\frac{1}{\sqrt N}$ in the compact form does not automatically mean the result is the formal uncertainty of the mean.

In formal metrology, NIST and the JCGM GUM describe the standard uncertainty of the mean as:

u(\bar x)=\frac{s_x}{\sqrt N}

For simple measurement practice, follow the convention used by the problem statement. If a problem uses $\Delta x$ for the compact formula above, call $\Delta x$ the repeated-measurement uncertainty for that problem model. The key is not to mix the two meanings without saying so.

When the Target Is Not the Diameter

Sometimes the measured data is diameter, but the required result is the area of the bottle cap. For a circle, the area is:

A=\frac{1}{4}\pi d^2

Because area depends on diameter, each diameter reading gives a slightly different area. Use $\pi \approx 3.14$ for this example calculation, and keep guard digits during the calculation.

Data	Diameter	Area from the diameter
$1$	$3.12 \text{ cm}$	$7.6415 \text{ cm}^2$
$2$	$3.14 \text{ cm}$	$7.7398 \text{ cm}^2$
$3$	$3.15 \text{ cm}$	$7.7892 \text{ cm}^2$
$4$	$3.11 \text{ cm}$	$7.5926 \text{ cm}^2$
$5$	$3.14 \text{ cm}$	$7.7398 \text{ cm}^2$

The mean area is:

\bar A=\frac{7.6415+7.7398+7.7892+7.5926+7.7398}{5}\text{ cm}^2=7.7006 \text{ cm}^2

The area spread is:

s_A=0.0807 \text{ cm}^2

The result can be reported as:

A=(7.70 \pm 0.08)\text{ cm}^2

Do not round the area column too early and then square the shortened values. Rounding in the middle of the calculation can change the final uncertainty.

A Percentage Shows the Size of the Risk

Relative uncertainty compares the uncertainty with the reported value.

\varepsilon_r=\frac{\Delta x}{|\bar x|}\times 100\%

For the bottle cap area:

\varepsilon_r=\frac{0.0807}{7.7006}\times 100\%=1.05\%

This means the spread of the data is about $1.05\%$ of the reported area. That is small enough for $7.70 \text{ cm}^2$ to be reasonable, but real enough that the uncertainty should still be written with the result.

The uncertainty references used here are NIST Technical Note $1297$ for Type A evaluation and the JCGM GUM document from BIPM. NIST's Type A evaluation page can be opened through nist.gov, while BIPM's JCGM GUM document can be opened through bipm.org.