Uncertainty in Repeated Measurements

Variation in Repeated Measurements

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 a number equal to or greater than zero that shows how widely measured values may be spread, based on the available data and information. For repeated measurements, a Type A evaluation calculates an uncertainty component from the observed results.

The sample standard deviation $s$ describes how far individual readings spread around their mean. The standard uncertainty of the mean describes the uncertainty in the mean estimated from those readings.

Repeated measurements show random variation through small changes between readings. Systematic errors, such as a tool that does not start at zero or a damaged scale, must be checked through inspection or calibration.

Mean of Repeated Measurements

The diameter of one bottle cap is measured $5$ times with a vernier caliper. Every reading uses the same unit, $\text{cm}$ .

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

For repeated data with a roughly symmetric spread and no clear outlier, the mean is an appropriate representative value.

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

For the data above:

Add the numerical readings in centimetres first:

\begin{aligned} S &=3.12+3.14+3.15 \\ &\quad{}+3.11+3.14 \\ &=15.66 \end{aligned}

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

Data 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 denominator $N-1$ is used because the sample mean $\bar d$ is calculated from the same data. The deviations from the mean must sum to zero, so only $N-1$ deviations can vary independently.

Sum the squared numerical deviations first:

\begin{aligned} V &=(-0.012)^2+(0.008)^2 \\ &\quad{}+(0.018)^2+(-0.022)^2 \\ &\quad{}+(0.008)^2 \\ &=0.00108 \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.

Data Spread and Uncertainty of the Mean

Some repeated-measurement exercises use the compact algebraic form below.

Abbreviate the two sums without changing the formula:

S_1=\sum_{i=1}^N x_i,\quad S_2=\sum_{i=1}^N x_i^2

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 a measurement problem, follow the notation defined in its statement. If $\Delta x$ denotes the compact formula above, then $\Delta x$ is the sample standard deviation of the individual measurements. Do not confuse this spread with the standard uncertainty of the mean $u(\bar x)=s_x/\sqrt N$ .

Area Calculated from Diameter Data

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 the calculator value of $\pi$ and keep the unrounded area values during the calculation. The table displays only $4$ decimal places for readability.

Data	Diameter	Area from the diameter
$1$	$3.12 \text{ cm}$	$7.6454 \text{ cm}^2$
$2$	$3.14 \text{ cm}$	$7.7437 \text{ cm}^2$
$3$	$3.15 \text{ cm}$	$7.7931 \text{ cm}^2$
$4$	$3.11 \text{ cm}$	$7.5964 \text{ cm}^2$
$5$	$3.14 \text{ cm}$	$7.7437 \text{ cm}^2$

The mean area is:

\bar A=\frac{1}{5}\sum_{i=1}^{5}A_i=7.7045 \text{ cm}^2

The area spread is:

s_A=0.0808 \text{ cm}^2

Using sample spread as the uncertainty convention for this exercise, the bottle-cap area can be written as follows.

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

The displayed table is not the calculation input. Rounding intermediate area values can change the final spread, so the mean and standard deviation above use the unrounded values.

Relative Spread as a Percentage of the Mean

Relative spread compares the sample standard deviation with the magnitude of the mean.

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

For the bottle cap area:

\varepsilon_r=\frac{0.0808}{7.7045}\times 100\,\%\approx 1.05\,\%

This means the sample spread is about $1.05\,\%$ of the mean area. Whether that is adequate depends on the intended use and target uncertainty, so the percentage should not be called "small enough" without a stated requirement.

Type A evaluation uses statistical analysis of repeated observations to estimate an uncertainty component. A reported measurement result should include enough information to state how its uncertainty was evaluated.