Lecture note #2, Physical Measurements, PHYS 201L


The word error in scientific context has a very restricted meaning.  It doesn’t mean to make a mistake.  Mistakes, such as measuring a 45.0 cm long table to be 35.0 cm, can be avoided by performing a very careful measurement.  Errors, on the other hand--cannot be avoided--even by the most careful observer.  It is a common practice amongst the scientist to divide the errors into two broad categories: systematic errors and random errors.


Systematic error:

This type of error is the result of an improperly calibrated apparatus or and improperly designed experiment that introduces the same one directional bias into all of the measurements.  A systematic error is an effect that changes all measurements by the same amount or by the same percentage.  For example, a ruler that has a badly worn one end, will introduce the same amount of uncertainty (in this case systematic errors) is introduced to all measurements.  Instrument zeroes should automatically be checked every time an instrument is used.


Random Error:

A random error is a result of fluctuations in experimental conditions (such as repetitive measurements) that cause a measured value to occur above or below the correct value with equal probability. For example, when we read the meter stick with naked eye in successive measurements, we may be unable to judge the position of the markings on the meter stick accurately enough to obtain repeatedly the same result.  This results in fluctuations in the measured values.  Sometimes, the fluctuations are intrinsic to the system under investigation (as in the radioactive source, where the number of the emerging radioactive particles arises from the basic nature of radioactive decay).  These uncertainties (or errors) can be estimated by using statistical methods (average, standard deviation, mean, mode,).  Note that there are other uncertainties, such as instrument uncertainty, which can be estimated by personal judgment.  For example, the instrument uncertainty of a meter stick is usually 0.1 cm. 


Instrument Uncertainty:

When an instrument is used in the laboratory, we should evaluate the uncertainty it introduces into the data collected.  It is ideal to assume that each instrument is calibrated against a known standard.  If this is the case the systematic errors are minimized.  If, on the other hand, this procedure is not possible, we can estimate the systematic errors by comparing the measurements of the same physical quantity taken with different instruments in the laboratory.  For instance, we can compare the measurements by using several meter sticks.  If they all agree within one millimeter (this also happens to be the smallest division), we can view this one-millimeter as the uncertainty with which our meter stick would agree when compared (or calibrated) to a standard meter.  Therefore the instrument uncertainty for the meter stick is ±0.1 cm. (± smallest division).  Sometimes, one can estimate the instrument uncertainty by interpolation.  The interpolation is usually estimated as a multiple of ˝, 1/3 or 1/5, etc of the smallest division on the instrument. In our lab, I recommend you use ˝ as the interpolation fraction.  If you use another multiple please indicate so in your report.

We can infer from the above discussion that,


Instrument uncertainty = interpolation fraction x smallest division.


When the data collected using an instrument is analyzed, the standard deviation of the repeated measurement (random error) should approximately equal the instrument uncertainty.



Accuracy implies the concept of correct answer or true value (accepted value) for a certain physical quantity, whereas precision refers to the reproducibility of a measurement.  Accuracy is a measure of how close the result of the experiment is to the true value.  Therefore, it is a measure of the correctness of the result.  The precision of an experiment is a measure of how well the result has been determined, without reference to its agreement with the true value.  An experiment that produces a value agreeing with the correct value is accurate.  An experiment that gives the same result (within some uncertainty) when repeated is precise.  An experiment with large random errors can be accurate but not precise.  An experiment with large systematic errors can be precise but will not be accurate.  While large systematic errors occur in one direction, therefore making the results inaccurate, in the same experiment, the outcome of the repeated measurements may yield same results implying high precision.