# 1.1: Measurement Error

- Page ID
- 34674

The result of a single measurement should be reported in the format \[(\textrm{estimate})\, \pm\, (\textrm{measurement error}).\] The **estimate** is your best guess for the true value, while the **measurement error** states the range where the true value might lie. By convention, the estimate and measurement error are formatted according to the these rules :

- The measurement error has
__one significant figure__. - The estimate has the same precision as the measurement error.

Suppose you use a digital multimeter to measure the current in a circuit, and the readout is stable (i.e., not fluctuating). Then you should report a result like this:

\(\;\;=\; \left(0.320 \pm 0.005\right) \, \mathrm{A}\)

Why? According to the readout, the value is between \(0.315\,\mathrm{A}\) (rounded up to \(0.32\,\mathrm{A}\)) and \(0.324999\dots\mathrm{A}\) (which is rounded down). So the measurement error is \(\pm 0.005 \, \mathrm{A}\). Note that the estimate is reported as \(0.320 \,\mathrm{A}\) to have the same precision as the error.

When using a device with hatch marks, such as a ruler or analog oscilloscope display, the measurement error is determined by the smallest markings. For example, if the smallest markings on a ruler have \(1\,\mathrm{mm}\) spacing, the measurement error is \(\pm 0.5 \,\textrm{mm}\), so a reading should be reported like this:

\(\;\;=\; \left(6.60 \pm 0.05\right) \, \mathrm{cm}\)

In more complicated situations, you must exercise your judgment. For instance, suppose you have a digital multimeter reading that is not stable: the last digit changes constantly, so that the reading fluctuates between \(0.32\), \(0.33\), and \(0.34\,\mathrm{A}\). The value is between \(0.315\,\mathrm{A}\) and \(0.344999\dots\mathrm{A}\), which is a range of \(\pm0.015\,\mathrm{A}\). Since we use one significant figure for errors, the result is reported like this:

\(\;\;=\; \left(0.33 \pm 0.02\right) \, \mathrm{A}\)

Alternatively, suppose the last digit is changing so fast that you can’t make out its values at all. Then you can report the result like this:

\(\;\;=\; \left(0.35 \pm 0.05\right) \, \mathrm{A}\)

Measurement uncertainties can also come from other aspects of an experiment. Suppose you use a ruler to measure the distance to an object, but the object wobbles by \(\pm 2\,\textrm{mm}\), larger than the \(1\,\textrm{mm}\) hatch marks of the ruler. In that case, you should report a measurement error of \(\pm 2\,\textrm{mm}\), not \(\pm 0.05\,\textrm{mm}\).