Physics · Ch 1 — Physical World and Measurement

Errors in Measurement

1.8

No measuring instrument, however well made, can determine a quantity with perfect exactness —

the limitations of the instrument, the observer, and the environment always leave some

uncertainty, called the error in the measurement. Understanding and estimating this

error is as important as recording the measurement itself, because a number reported without

any sense of its uncertainty cannot be properly compared against a theoretical prediction or

another experiment's result.

\text{[see table: Types of measurement error]}

Systematic errors are the most dangerous because they are consistent, not random — they

bias every single reading in the same direction, so simply averaging more readings does

not remove them. Common sources include: an instrument with a zero error (it does not

read exactly zero when it should), an incorrectly calibrated scale, or an experimental

procedure with a built-in bias (e.g. always starting a stopwatch a fraction of a second late).

Systematic errors must be identified and corrected, e.g. by applying a zero correction.

Random errors arise from small, unpredictable fluctuations — a slightly different reaction

time when starting/stopping a stopwatch, tiny air currents disturbing a balance, small

parallax differences in reading a scale. Random errors scatter the readings both above and

below the true value, so — unlike systematic errors — averaging a large number of repeated

readings genuinely reduces their effect on the reported mean.

Gross errors are outright blunders: misreading an instrument, recording the wrong digit,

or using the wrong formula. These are avoided through careful, repeated observation rather

than any mathematical correction.

For a set of $n$ repeated readings $a_1, a_2, \ldots, a_n$ of the same quantity:

\bar a = \frac{a_1+a_2+\cdots+a_n}{n}\ \text{(the mean, taken as the best estimate)}

\Delta a_i = |\bar a - a_i|\ \text{(the absolute error of the $i$-th reading)}

\Delta a_{\text{mean}} = \frac{\Delta a_1 + \Delta a_2 + \cdots + \Delta a_n}{n}\ \text{(mean absolute error)}

\text{Percentage error} = \frac{\Delta a_{\text{mean}}}{\bar a}\times 100\%

Table 3Types of measurement error

Type of Error	Cause	Typical Example	Can it be reduced?
Systematic error	A flaw that biases every reading the same way	Zero error in a screw gauge; a mis-calibrated scale	Yes — by correcting the instrument / applying a zero correction
Random error	Unpredictable small fluctuations from reading to reading	Slight change in the observer's reaction time while using a stopwatch	Reduced (not eliminated) by averaging many readings