Accuracy and Precision
Like all the physical sciences, physics is based on the results of measurements. Every measurement has
uncertainty associated with it. The amount of uncertainty is related to the measuring devices
themselves and to other factors. To reduce all types of error, one should use the same measuring
device throughout an experiment.
Two terms are usually applied in discussing uncertainty in measured quantities: accuracy and precision.
Accuracy is a measure of how closely one measurement agrees with an established value. This value
should be from a reliable printed source or from a well-documented website. Accuracy depends on the
calibration of the measuring device. If the instrument is not calibrated correctly (flawed), then the
measured value will be inaccurate. It also depends on the quality of the measuring device.
Precision is a measure of how closely a set of measurements are to one another. The precision provides
a gauge as to how reliable the experimental procedure is or how reproducible it is. Precision depends
on the skill of the person using the measuring device.
There are two types of errors that may be experienced: systematic (determinate) and random
(indeterminate). If an experiment contains a systematic error, then increasing the sample size will
increase the precision but will not affect the accuracy. A systematic error causes the value of the
measurement to be off in the same direction each time. Eliminating the error will improve the accuracy
but will not change the precision. The presence of an interfering substance, incorrect calibration,
method error, and operator error are all types of systematic errors that affect the accuracy of a
measurement. Systematic errors can be reduced by careful adjustment of the experimental procedure
and by careful calibration of equipment. The amount of systematic error may be expressed by using
percent error.
A random error means that a measurement may be high or low. This error occurs in the estimating of
the last digit. Some random errors are inherent in the procedure and some are just the nature of the
concept being studied. If there is a random error then the precision will be poor but one or more
samples may be accurate. Random errors may be minimized but cannot not be removed. Random
errors can be minimized by increasing the sample size or the number of samples.
There is a way of expressing experimental results so as to obtain a realistic value for the measurement.
First find the mean value of the measurements. Then find the deviation of each measurement from the
mean by subtraction. Lastly find the average deviation. The measurement is then expressed as the
mean value ± average deviation. This gives a range of possible values for the measurement which
incorporates the precision of the instrument used.
One may calculate percent error and percent deviation.
A common method of conveying accuracy and precision is to use significant figures. Significant figures
are all those digits which may be read from an instrument plus one digit which is estimated. Each
instrument should be checked carefully before using it to determine how many figures may be read
Accuracy and Precision
from it. Then estimate one more figure. The use of significant figures lends reliability and consistency
to measurements.
What digits are significant?
All nonzero digits are significant.
All captive zeroes are significant.
All beginning zeroes are NOT significant.
All zeroes to the right of a nonzero digit and a decimal point are significant.
Ending zeroes in numbers without a decimal may or may not be significant. The final
significant zero may be noted with a bar over it.
Remember that numbers obtained by counting, such as the number of people in a car, and the
numbers in definitions are exact numbers and, hence, have an infinite number of significant
figures.
What about calculations?
When adding or subtracting, the answer may have only as many DECIMAL PLACES as the least
precise number in the operation.
When multiplying or dividing, the answer may have only as many SIGNIFICANT DIGITS as the
least precise number in the operation.
In pH, which is a logarithmic value, only the digits to the right of the decimal point are
significant. The digit to the left of the decimal indicates a place value. In other words, pH =
10.26 has only two significant figures and it corresponds to a [H+] = 5.5 x 10-11 M which has two
significant figures.