Random or Statistical Uncertainties

Random or statistical uncertainties arise from random fluctuations in a measurement. These random fluctuations can occur in measuring devices. For example, electronic noise and air currents lead to a rapid but small fluctuation in motion detector readings. These fluctuations occur, even when the motion detector is measuring the distance to a stationary object. Random fluctuations can also be a characteristic of the quantity being measured. For example, if we use a meter stick to measure the landing positions of a series of projectiles shot from a spring-loaded launcher, we see significant random variations which clearly do not arise from the limitations of the meter stick. Instead, we suspect that the launch velocity given to projectiles by the launcher is subject to small random variations.

Truly random fluctuations average to zero, and so the way to remove them is to average a large number of measurements,

\bar{x} = \frac{1}{N} \sum_{i=1}^N x_i
\end{displaymath} (1)

The average value $\bar{x}$ approaches the ``true value'' as the number of measurements in the average approaches infinity. Finding the ``true value'' is impractical, so we settle for the ``best value'' given by the average. The average value is also called the mean value.

Random fluctuations are described by the normal distribution, or Gaussian distribution, or the ``bell curve.'' The uncertainty in the ``best value'' of a large collection of normally distributed measurements can be calculated using the standard deviation

\sigma_x = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x})^2}
\end{displaymath} (2)

which describes the width of the distribution. More precisely, about 68% of a normal distribution falls within $\sigma$ of the average value. The standard deviation is the uncertainty in a single measurement in the distribution. Rather than doing this calculation ``by hand,'' I recommend using the STDEV() function of your spreadsheet.

The uncertainty in the average of a large number of measurements is less than $\sigma$. This follows from the idea that the more measurements we make, the closer the average value comes to the ``true value.'' The standard deviation of the mean is given by

\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{N}}
\end{displaymath} (3)

We report this as the uncertainty in $\bar{x}$.

See the sample write-up in Appendix A for an example of an analysis of normally distributed data.

Systematic Errors

Systematic errors are are due to a defect in the equipment or methods used to make measurements. For example, a motion sensor can be poorly calibrated so that it gives distance readings which are only 90% of the true values. It has a systematic uncertainty (10%) that is much greater in magnitude than the statistical uncertainty in its readings. Systematic errors are often difficult to detect, because they do not show up as fluctuations in the results of repeated measurements. It is important to think about possible sources of systematic errors and to try to correct them or rule them out, for example by

Estimating Uncertainties

We often do not have the luxury of a large collection of normally distributed measurements to analyze. Instead, we must somehow estimate the uncertainty of a single measurement. This is necessarily somewhat subjective. If only a few measurements are available, it is more reasonable to use the entire range covered by the measurements to define the uncertainty instead of calculating the standard deviation of the mean. If only one measurement is available the resolution of the device and the variation in the quantity measured are important guides. For example, the resolution of a meter stick is 1 mm. If it is used to measure the length of a rectangular steel plate, an uncertainty of 1 mm, or perhaps even 0.5 mm, is reasonable. If I use the same meter stick to measure the height of a small child, issues of variable posture and how I line up the stick lead to an uncertainty of as much as 1 cm. Be conservative with your estimates. That is, when in doubt, it is a good policy to report a larger uncertainty.

Propagation of Uncertainties in Calculations

Frequently, calculations involve one or more measured quantity, and we need to determine how the uncertainties in input quantities translate into the uncertainty in the result. The guidelines below cover all of the possibilities. Always check that the result and its corresponding uncertainty have the same units. If they do not, something went wrong.


When adding or subtracting, add absolute uncertainties in quadrature.

For example, if $d = a+b-c$, then

\sigma_d = \sqrt{\sigma_a^2+\sigma_b^2+\sigma_c^2}
\end{displaymath} (4)


When multiplying or dividing, add relative (percentage) uncertainties in quadrature.

For example, if $d = \frac{ab}{c}$, then

\frac{\sigma_d}{d} = \sqrt{
\left( \frac{\sigma_a}{a} \right...
...{\sigma_b}{b} \right)^2
+\left( \frac{\sigma_c}{c} \right)^2
\end{displaymath} (5)

When raising a value to a power, multiply its relative error by the power. For example, if $d = \frac{a^lb^m}{c^n}$
\frac{\sigma_d}{d} = \sqrt{
\left(l \frac{\sigma_a}{a} \righ...
...\sigma_b}{b} \right)^2
+\left(n \frac{\sigma_c}{c} \right)^2
\end{displaymath} (6)

In General (Approximately)

Use first derivatives to determine the approximate variation of the result due to the uncertainty in each measured quantity.

If a quantity $f$ is a function of the measured quantities $a, b, c,
...$, then

\sigma_f = \sqrt{
\left(\frac{\partial f}{\partial a} \right...
...ft(\frac{\partial f}{\partial c} \right)^2 \sigma_c^2
+ ...
\end{displaymath} (7)

In General (Exact)

When calculating a result which depends on measured input quantities, determine the variations in the result due to each input quantity, and add the variations in quadrature. In some cases, upper and lower uncertainties differ.

For example, if $d = a \log b$, the individual variances are

$\displaystyle \sigma_{da+}$ $\textstyle =$ $\displaystyle \vert(a+\sigma_a) \log b - d\vert$  
$\displaystyle \sigma_{da-}$ $\textstyle =$ $\displaystyle \vert(a-\sigma_a) \log b - d\vert$  
$\displaystyle \sigma_{db+}$ $\textstyle =$ $\displaystyle \vert a \log (b+\sigma_b) - d\vert$  
$\displaystyle \sigma_{db-}$ $\textstyle =$ $\displaystyle \vert a \log (b-\sigma_b) - d\vert$ (8)

and the upper and lower uncertainties are
$\displaystyle \sigma_{d+}$ $\textstyle =$ $\displaystyle \sqrt{\sigma_{da+}^2 + \sigma_{db+}^2}$  
$\displaystyle \sigma_{d-}$ $\textstyle =$ $\displaystyle \sqrt{\sigma_{da-}^2 + \sigma_{db-}^2}$ (9)

This kind of analysis is a good job for a spreadsheet.

Reporting Results with Uncertainties

Results with uncertainties are typically reported in the form
x \pm \sigma_x
\end{displaymath} (10)

Units are always included, and are usually given after the result and its uncertainty. It is common practice to round uncertainties to one significant figure. Results should be rounded off to the decimal place of the corresponding uncertainties. For example, if an analysis of several measurements of my height reveals an average of $\bar{h} = 1.8037$ m with a standard deviation of the mean of $\sigma_{\bar{h}} = 0.00566$ m, I report my height as $h = 1.804 \pm
0.006$ m. The form
\end{displaymath} (11)

is also sometimes used, where the uncertainty is given as a single digit. In this form, my height is $h = 1.804(6)$ m. The uncertainty is assumed to be in the last reported digit of the result. With asymmetric uncertainties, one uses the form
\end{displaymath} (12)

Copyright © 2002-2004, Lewis A. Riley Updated Mon Jan 19 13:29:10 2004

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