Precision, Accuracy, and Significant Figures
Accuracy:
The measurement of a physical quantity is always subject to some degree of uncertainty. There are several reasons for this: the
limitations inherent in the construction of the measuring instrument or device, the conditions under which the measurement was
made, and the different ways in which the person uses or reads the instrument. Consequently, in reporting the measurements
made during a scientific experiment, it is necessary to indicate the degree of uncertainty so far as it is known.
One way to express the uncertainty of a measurement is in terms of accuracy. Accuracy refers to the closeness of a measurement
to the accepted value for a specific physical quantity. It is expressed as either an absolute or relative error. Absolute error is the
actual difference between the measured value and the accepted value. The equation for absolute error is:
, where Ea is the absolute error, O is the observed (measured) value, and A is the accepted value. ('The vertical
bars mean that only the absolute value is used, regardless of sign, thus it is always a positive number) Relative error is expressed
as a percentage, and it is therefore often called the percentage error. It is calculated as follows:
where Er is the relative error, Ea is the absolute error, and A the accepted value.
In common usage, accuracy and precision are frequently synonymous terms. But in science it is important to make a distinction
between them. You should learn to use them correctly and consistently
Precision:
In contrast with accuracy, precision is the agreement among several measurements that have been made in the same way. It tells
how reproducible the measurements are and is expressed in terms of deviation. As in the case of accuracy, deviations in
precision can be given as absolute or relative
(percentage).
Absolute deviation is the difference between a single measured value and the average of several measurements made in the same
way, or
where Da is the absolute deviation, O is the observed or measured value, and M is the mean or average of several readings.
Relative deviation is the percentage average deviation of a set of measurements and is calculated as follows:
where Dr, is the relative deviation, Da is the absolute deviation, and M is the mean or average of` the set of
readings. Since relative deviation is based upon percent, it gives a more realistic description of precision
than does the absolute deviation.
The precision of your laboratory measurements will be governed by the instruments at your disposal. In
a measuring instrument, the degree of precision obtainable is called the tolerance of the device. Any
figure for the tolerance of an instrument indicates only the limitations of the instrument. The
instrument maker assumes that the instrument is used properly and that human errors are held to a
minimum. A metric ruler, for example, has a tolerance of plus or minus 0.01 cm. Some modern research
instruments are almost unbelievable precise. However, all measuring instruments have some degree of
uncertainty associated with them! This is why we must pay attention to significant figures when
recording measurements and when doing calculations based upon measurements.
Significant Figures:
All measurements are composed of some digits which are known with certainty, plus one uncertain digit. Because of this, we
apply the rules of significant figures when recording measured values and when conducting calculations based upon those
values.
Recording measurements: Always record measurements based upon the limitations of your measuring
device; no more, no less!!!
Example: A meter stick, which has graduations to the nearest 0.1 cm (or 1 mm) should allow you
to record a measurement to the nearest 0.01 cm (or 0.1 mm). You can always record a value one
“place value” beyond the calibrated marks on your measuring device.
In the example below, the length of the Dark line should be recorded as 3.34 cm. Note that the
last digit is uncertain. While another student could record it as 3.33 cm or 3.35 cm, the 3.3 is
certain, while the value in the hundredths place is uncertain.
Calculations involving significant figures: A calculation involving measured quantities is recorded to
express the same precision as the least precise measurement. This is usually accomplished by applying the
following rule:
1.
The sum or difference of two or more numbers should contain as many decimal as the
measurement with the smallest number of decimal places. If an integer is involved in addition or
subtraction, the sum or the difference should contain no decimal places unless the integer is an
exact number.
2. The product or quotient of two or more should be recorded with as many significant figures as
the least precise measurement used to arrive at that product or quotient.
3. Note: When performing a series of calculations, it is best to “carry” as many digits as possible
throughout your calculations, then round appropriately at the end.