All science is concerned with
measurement. This fact requires that we have standards of measurement.
In order to make meaningful
measurements in science we need standards of commonly measured
quantities, such as those of mass, length and time. These standards are
The kilogram is the mass of a cylinder of platinum-iridium alloy
kept at the International Bureau of Weights and Measures in Paris.
metre is defined as the length of the path travelled by light in a
vacuum during a time interval of 1/299 792 458 of a second. (Note that
the effect of this definition is to fix the speed of light in a vacuum
at exactly 299 792 458 m·s-1).
second is the duration of 9 192 631 770 periods of the radiation
corresponding to the transition between the two hyperfine levels of the
ground state of the caesium 133 atom.
It is necessary for all such
standards to be constant, accessible and easily reproducible.
Scientists all over the world
use the same system of units to measure physical quantities. This
system is the International System of Units, universally abbreviated SI
(from the French Le Système International d'Unités). This is the
modern metric system of measurement. The SI was established in 1960 by
the 11th General Conference on Weights and Measures (CGPM, Conférence
Générale des Poids et Mesures). The CGPM is the international
authority that ensures wide dissemination of the SI and modifies the SI
as necessary to reflect the latest advances in science and technology.
Thus, the kilogram, metre and
second are the SI units of mass, length and time respectively. They are
abbreviated as kg, m and s. Various prefixes are used to help express
the size of quantities – eg a nanometre = 10-9 of a metre; a
gigametre = 109 metres. See the table of prefixes below.
Table 1. SI prefixes
Fundamental and Derived Quantities
Physical quantities are not
generally independent of one another. Many quantities can be expressed
in terms of more fundamental quantities. The first three fundamental
quantities we will deal with are those of mass, length and time. Many
derived quantities can be expressed in terms of these three. For
example, the derived quantity speed can be expressed as length/time.
Note that there are seven
fundamental quantities in all. The other four are: current,
thermodynamic temperature, amount of substance and luminous intensity.
We will deal with these as we need them.
The expression of a derived
quantity in terms of fundamental quantities is called the dimension of
the derived quantity. The symbol M is used to denote the dimension of
mass, as is L for length and T for time.
So, for example, to determine
the dimensions of the derived quantity speed, we would look at the
formula for speed, namely:
speed = distance/time
The dimensions of speed are
[speed] = [distance]
/ [time] = L / T
where the square brackets are
used to indicate that we are calculating the dimensions of whatever is
inside the brackets.
Finally, we use our knowledge of
indices to simplify this expression.
[speed] = LT-1
Question: Determine the dimensions of (a) area and (b)
Answers: (a) L2; (b) L3.
Dimensions can be used to check
the correctness of an equation. The dimensions of the left hand side of
the equation must equal the dimensions of the right hand side.
Dimensions can also be used to verify that different mathematical
expressions for a given quantity are equivalent.
Given the formulas for the following derived quantities, calculate the
dimensions of each quantity.
velocity = displacement/time
acceleration = change of velocity/time
momentum = mass x velocity
force = mass x acceleration
work = force x displacement
a. LT-1; b. LT-2; c. MLT-1; d. M LT-2;
e. M L2T-2.
Since the precision of all
measuring instruments is limited, the number of digits that can be
assumed as known for any measurement is also limited. When making a
measurement, read the instrument to its smallest scale division.
Estimate within a part of a division. The figures you write down for
the measurement are called significant figures.
In Physics, if you write 3.0,
you are stating that you were able to estimate the first decimal place
of the quantity and you are implying an error of 0.05 units. If you
just write 3, you are stating that you were unable to determine the
first decimal place and you are implying an error of 0.5 units.
It is very important that you do
not overstate the precision of a measurement or of a calculated
quantity. A calculated quantity cannot have more significant figures
than the measurements or supplied data used in the calculation. So,
for example, if the length, breadth & height of a rectangular prism is
each known to 2 significant figures, the volume calculated from these
figures cannot have more than 2 significant figures. Let’s say the
volume = 3.7cm x 2.9cm x 5.1cm = 54.723 cm3. You would state
the volume as 55cm3 (2 significant figures only). Note that
we have rounded the volume up to the nearest whole number in this case.
Zeros between the decimal point and the
first non-zero digit are not significant. eg 0.00035 has 2
Zeros that round off a large number
are not significant. eg 35,000 has 2 significant figures.
Zeros at the end of a string of
decimals are significant. eg 0.5500 has 4 significant figures.
The last 2 digits are meaningful here. The measurement is 0.5500 not
0.5501 or 0.5499.
Zeros in between non-zero digits are
significant. eg 0.7001 has 4 significant figures. The first zero is
not significant but the next two are.
Order & Scientific Notation
The order of a number is the
nearest power of 10 to that number. eg 166,000 has an order of 105;
756,000 has an order of 106; 0.099 has an order of 10-1.
In Physics quite often
scientific notation is used. Write one non-zero figure before the
decimal point and correct the magnitude of the number by using the
appropriate power of ten. eg 166,000 can be written as 1.66 x 105;
0.099 can be written as 9.9 x 10-2.
RELIABILITY AND VALIDITY
These three terms are often used when referring to experiments, experimental
results and data sources in Science. It is very important that students
have a good understanding of the meaning and use of these terms.
The following notes under the blue headings were taken from “Optimizing
Student Engagement and Results in the Quanta to Quarks Option” by
Dr Mark Butler, Gosford High School. The full
article may be found at the link below.
a) ACCURACY: Conformity to truth.
texts refer to accuracy in two ways:
Accuracy of a result or experimental procedure can refer to the
percentage difference between the experimental result and the accepted
value. The stated uncertainty in an experimental result should always be
greater than this percentage accuracy.
Accuracy is also associated with the inherent uncertainty in a
measurement. We can express the accuracy of a measurement explicitly by
stating the estimated uncertainty or implicitly by the number of
significant figures given. For example, we can measure a small distance
with poor accuracy using a metre rule, or with much greater accuracy
using a micrometer. Accurate measurements do not ensure an experiment is
valid or reliable. For example consider an experiment for finding g in
which the time for a piece of paper to fall once to the floor is
measured very accurately. Clearly this experiment would not be valid or
reliable (unless it was carried out in vacuum).
b) RELIABILITY: Trustworthy, dependable.
terms of first hand investigations reliability
can be defined as repeatability or consistency. If an experiment is repeated many times
it will give identical results if it is reliable. In terms of second
hand sources reliability refers to how trustworthy the source is. For
example the NASA web site would be a more reliable source than a private
web page. (This is not to say that all the data on the site is valid.)
The reliability of a site can be assessed by comparing it to several
Derived correctly from premises already accepted, sound, supported by
valid experiment is one that fairly tests the hypothesis. In a
valid experiment all variables are kept constant apart from those being
investigated, all systematic errors have been eliminated and random
errors are reduced by taking the mean of multiple measurements. An
experiment could produce reliable results but be invalid (for example
Millikan consistently got the wrong value for the charge of the electron
because he was working with the wrong coefficient of viscosity for air).
An unreliable experiment must be inaccurate, and invalid as a valid
scientific experiment would produce reliable results in multiple trials.
- The notes below on accuracy & precision, nature & use of errors
and determination of errors are my own work.
ACCURACY & PRECISION
Another term you will hear in relation to experiments and experimental
results is the term precision. Precision is the degree of
exactness with which a quantity is measured. It refers to the
repeatability of the measurement. The term precision is
therefore interchangeable with the term reliability. The two terms
mean the same thing but you will hear & read both in relation to science
experiments & experimental results.
The precision of a measuring device is limited by the finest division on
Note too, that a highly precise measurement is not necessarily an
accurate one. As indicated in the first definition of accuracy
above, accuracy is the extent to which a measured value agrees
with the "true" or accepted value for a quantity. In scientific
experiments, we aim to obtain results that are both accurate and
precise. The section on errors below will hopefully further
clarify the four important terms defined in these last two sections of
notes - accuracy, reliability, precision & validity.
USE OF ERRORS
Errors occur in all physical measurements.
When a measurement is used in a calculation, the error in the
measurement is therefore carried through into the result. The two different types of error that can
occur in a measured value are:
Systematic error – this occurs to the same
extent in each one of a series of measurements eg zero error, where for
instance the needle of a voltmeter is not correctly adjusted to read
zero when no voltage is present.
Random error – this occurs in any
measurement as a result of variations in the measurement technique (eg
parallax error, limit of reading, etc).
When we report errors in a measured quantity we
give either the absolute error, which is the actual size of the
error expressed in the appropriate units or the relative error,
which is the absolute error expressed as a fraction of the actual
measured quantity. Relative
errors can also be expressed as percentage errors.
So, for instance, we may have measured the acceleration due to
gravity as 9.8 m/s2 and determined the error to be 0.2 m/s2. So, we say the absolute error in the result is 0.2 m/s2
and the relative error is 0.2 / 9.8 = 0.02 (or 2%).
Note relative errors have no units.
We would then say that our experimentally determined value for
the acceleration due to gravity is in error by 2% and therefore lies
somewhere between 9.8 – 0.2 = 9.6 m/s2 and 9.8 + 0.2 = 10.0
m/s2. So we
write g = 9.8 ±
0.2 m/s2. Note
that determination of errors is beyond the scope of the current course.
Consider three experimental determinations of g,
the acceleration due to gravity.
We can say that Experiment
A is more reliable (or precise) than Experiment
B because its relative error is smaller and therefore if the
experiment was repeated we would be likely to get a value for g
which is very close to the one already obtained.
That is, Experiment A has
results that are very repeatable (reproducible).
Experiment B, however, is
much more accurate than Experiment A,
since its value of g is much closer to the accepted value.
Clearly, Experiment C is
neither accurate nor reliable.
In terms of validity, we could say that Experiment
B is quite valid since its result is very accurate and
reasonably reliable – repeating the experiment would obtain reasonably
similar results. Experiment
A is not valid, since its result is inaccurate and Experiment
C is invalid since it is both inaccurate and unreliable.
How do you improve the reliability of an
experiment? Clearly, you
need to make the experimental results highly reproducible.
You need to reduce the relative error (or spread) in the
results as much as possible. To
do this you must reduce the random errors by: (i) using
appropriate measuring instruments in the correct manner (eg use a
micrometer screw gauge rather than a metre ruler to measure the diameter
of a small ball bearing); and (ii) taking the mean of multiple
To improve the accuracy and validity of an
experiment you need to keep all variables constant other than those
being investigated, you must eliminate all systematic errors by careful
planning and performance of the experiment and you must reduce random
errors as much as possible by taking the mean of multiple measurements.
DETERMINATION OF ERRORS
All experimental science
involves the measurement of quantities and the reporting of those
measurements to other people. We have already seen that stating the
absolute and relative errors in our measurements allows people to decide
the degree to which our experimental results are reliable. This in turn
helps people to decide whether our results are valid or not.
Clearly then it is important for
all scientists to understand the nature and sources of errors and to
understand how to calculate errors in quantities. A whole branch of
mathematics has been devoted to error theory. Methods exist to estimate
the size of the error in a result, calculated from any number of
measurements, using any combination of mathematical operations. We will
investigate a few of these methods appropriate for high school Physics
Variations will occur in any
series of measurements taken with a suitably sensitive measuring
instrument. The variations in different readings of a measurement are
usually referred to as “experimental errors”. They are not to be
confused with “mistakes”. Such variations are normal.
Let’s say we use a micrometer
screw gauge to measure the diameter of a piece of copper wire. The
micrometer allows us to read down to 0.01mm. We may obtain a set of
readings in mm such as: 0.73, 0.71, 0.75, 0.71, 0.70, 0.72, 0.74, 0.73,
0.71 and 0.73.
The variation in these figures
is probably mainly due to the fact that the wire is not of uniform
diameter along its length. However, the variation could also be caused
by slight variations in the measuring technique – closing the jaws of
the micrometer more or less tightly from one measurement to the next.
The experimenter may have occasionally read the scale at an angle other
than perpendicular to the scale, thus introducing parallax error into
the results. Such factors as these cause random variations in the
measurements and are therefore called Random Errors. The
question we must ask is: How do we take account of the effects of
random errors in analysing and reporting our experimental results?
If we had taken say 50 readings
of the diameter of the wire instead of just 10, we could use our
knowledge of Statistics to draw a frequency histogram of our
measurements, showing the number of times each particular value occurs.
This would be very helpful to anyone reading our results since at a
glance they could then see the nature of the distribution of our
readings. If the number of readings we take is very high, so that a
fine subdivision of the scale of readings can be made, the histogram
approaches a continuous curve and this is called a distribution curve.
If the errors are truly random,
the particular distribution curve we will get is the bell-shaped
Normal (or Gaussian) Distribution shown below.
The readings or measured values
of a quantity lie along the x-axis and the frequencies (number of
occurrences) of the measured values lie along the y-axis. The Normal
Curve is a smooth, continuous curve and is symmetrical about a central
“x” value. The peak in frequency occurs at this central x value.
The basic idea here is that if
we could make an infinite number of readings of a quantity and graph the
frequencies of readings versus the readings themselves, random errors
would produce as many readings above the “actual” or “true” value
of the quantity as below the “true” value and the graph that we would
end up with is the Normal Curve. The value that occurs at the
centre of the Normal Curve, called the mean of the normal
distribution, can then be taken as a very good estimate of the “true”
value of a measured quantity.
So, we can start to answer the
question we asked above. The effect of random errors on a
measurement of a quantity can be largely nullified by taking a large
number of readings and finding their mean. The formula for the
mean is, of course, as shown below:
Examine the set of micrometer
readings we had for the diameter of the copper wire. Let us calculate
their mean, the deviation of each reading from the mean and the squares
of the deviations from the mean.
Squares of Deviations
(mm) From Mean
+ 0.01 0.0001
- 0.01 0.0001
+ 0.03 0.0009
- 0.01 0.0001
- 0.02 0.0004
+ 0.02 0.0004
+ 0.01 0.0001
- 0.01 0.0001
+ 0.01 0.0001
For the moment we will only be
interested in the first two columns above. A glance at the deviations
shows the random nature of the scattering.
The formula for the mean yields:
The mean is calculated as 0.723
mm but since there are only two significant figures in the readings, we
can only allow two significant figures in the mean. So, the mean is
0.72 mm. Once we have the mean, we can calculate the figures in the 2nd
column of the Table above. These are the deviation of each reading from
We can use the maximum deviation
from the mean, 0.03 mm, as the “maximum probable error (MPE)” in
the diameter measurements. So, we can state the diameter of the copper
wire as 0.72 ± 0.03 mm (a 4% error). This means that the
diameter lies between 0.69 mm and 0.75mm.
An interesting thought occurs:
What if all the readings of the diameter of the wire had worked out to
be the same? What would we use as an estimate of the error then?
In that case, we would look at
the limit of reading of the measuring instrument and use half of
that limit as an estimate of the probable error. So, as stated above,
our micrometer screw gauge had a limit of reading of 0.01mm. Half the
limit of reading is therefore 0.005mm. The diameter would then be
reported as 0.72 ± 0.005 mm (a 0.7% error). This means that the
diameter lies between 0.715 mm and 0.725 mm. Note that we still only
quote a maximum of two significant figures in reporting the diameter.
It is also worth emphasizing
that in the stated value of any measurement only the last digit
should be subject to error. For example, you would not state the
diameter of the wire above as 0.723 ± 0.030 mm because the error is in
the 2nd decimal place. This makes the 3rd decimal
place meaningless. If you do not know the 2nd decimal place
for certain, there is no point stating a 3rd decimal place in
the value of the quantity. So, do not write an answer to 5 decimal
places just because your calculator says so. Think about how many
figures are really significant.
We can now complete our answer
to the question: How do we take account of the effects of random
errors in analysing and reporting our experimental results? At high
school level, it is sufficient to:
Take a large number of readings – at
least 10, where time and practicality permit.
Calculate the mean of the readings as a
reasonable estimate of the “true” value of the quantity.
Use the largest deviation of any of the
readings from the mean as the maximum probable error in the mean value.
If all the readings are the same, use
half the limit of reading of the measuring instrument as the MPE in the
Now, for those who would like to
go a little further in error theory, we can turn our attention to the
third column of figures in the Table above. These figures are the
squares of the deviations from the mean. Without going into any
theoretical explanation, it is common practice for scientists to use a
quantity called the sample standard deviation of a set of
readings as an estimate of the error in a measured quantity. The
case sigma), is calculated from the squares of the deviations from
the mean using the following formula:
From the 3rd column
above we have
n (the number of readings) = 10
therefore report the diameter of the copper wire as 0.72 ±
0.016 mm (a 2% error). This means that the diameter lies between
0.704 mm and 0.736 mm. Note that we still only quote a maximum of two
significant figures in reporting the diameter.
Why do scientists use standard
deviation as an estimate of the error in a measured quantity? Well, the
standard deviation of a set of experimental data is a reliable
statistical measure of the variability or spread of the data from the
mean. A high standard deviation indicates that the data is spread out
over a large range of values, whereas a low standard deviation indicates
that the data values tend to be very close to the mean.
Also, standard deviation gives us a
measure of the percentage of data values that lie within set distances
from the mean. If a data distribution is approximately normal then
about 68% of the data values are within 1 standard deviation of the mean
(mathematically, μ ± σ, where μ is the arithmetic mean), about 95% are
within two standard deviations (μ ± 2σ), and about 99.7% lie within 3
standard deviations (μ ± 3σ). So, when we quote the standard deviation
as an estimate of the error in a measured quantity, we know that our
error range around our mean (“true”) value covers the majority of our
data values. In other words, it can give us a level of confidence in
our error estimate. If you wish, you could quote the error estimate as
two standard deviations.
Systematic errors are
errors which occur to the same extent in each one of a series of
measurements. Causes of systematic error include:
Using the instrument wrongly on a
consistent basis. A simple example is parallax error, where
you view the scale of a measuring instrument at an angle rather than
from directly in front of it (ie perpendicular to it). If this is done
consistently, it introduces a systematic error into the results. A
person sitting in the passenger seat of a car for instance may glance at
the speedometer and think the driver is going above the speed limit by a
couple of km/hr, when in fact the driver, sitting directly in front of
the speedometer, can see that the speed of the car is right on the speed
The instrument may have a built in
error. A simple example is zero error, where the instrument
has not been correctly set to zero before commencing the measuring
procedure. An ammeter for instance may show a reading of 0.2A when no
current is flowing. So, as you use the instrument to measure various
currents each of your measurements will be in error by 0.2A. The
ammeter needle should have been reset to zero by using the adjusting
screw before the measurements were taken.
External conditions can introduce
systematic errors. A metal rule calibrated for use at 25oC
will only be accurate at that temperature. If you use this rule say at
5oC it will produce readings that are consistently larger
than they should be since at the lower temperature the metal will have
contracted and the distance between each scale division will be smaller
than it should be. Knowing the expansion coefficient of the metal would
allow the experimenter to correct for this error.
Systematic errors can
drastically affect the accuracy of a set of measurements. Unfortunately,
systematic errors often remain hidden. Clearly, to reduce the incidence
of systematic errors the experimenter must:
Use all measuring instruments correctly
and under the appropriate conditions.
Check for zero error. This can include
performing test measurements where a standard or known quantity is
measured to ensure that the instrument is giving accurate results. For
example, a thermometer could be checked at the temperatures of melting
ice and steam at 1 atmosphere pressure.
Errors in Calculated
In scientific experiments we
often use the measured values of particular quantities to calculate a
new quantity. The error in the new quantity depends on the errors in
the measured values used to calculate it.
Addition & Subtraction
When two (or more) quantities
are added or subtracted to calculate a new quantity, we add the maximum
probable errors in each quantity to obtain the maximum probable error in
the new quantity.
When heating water we may measure the starting temperature to be (35.0
and the final temperature to be (85 ± 0.5)oC.
The change in temperature is therefore (85.0 – 35.0)oC
or (50.0 ± 1.0)oC.
Note that we add the MPE’s in the measurements to obtain the MPE in the
Multiplication & Division
When two (or more) quantities
are multiplied or divided to calculate a new quantity, we add the
percentage errors in each quantity to obtain the percentage error in the
For Example: Let us assume we are to
determine the volume of a spherical ball bearing. After performing a
series of measurements of the radius using a micrometer screw gauge, the
mean value of the radius is found to be 9.53mm ±
0.05mm. Thus, the percentage error in the radius is 0.5%. [ % error =
The formula for the volume of a
V = 4/3
Using this formula, the value
for the volume of the ball bearing is found to be 3625.50mm3.
Note that the only measured
quantity used in this calculation is the radius but it appears raised to
the power of 3. The formula is really:
x r x r
So, % error in
volume = % error in r + % error in r + % error in r
error in volume = 0.5% + 0.5% + 0.5% = 1.5%
The volume of the
ball bearing is therefore 3625.50mm3
± 1.5% or
3625.50mm3 ± 54.38mm3.
Now we look at the number of
significant figures to check that we have not overstated our level of
precision. There are only 3 significant figures in the radius
measurement. We should therefore have only 3 significant figures in the
volume. Writing the volume figure in more appropriate units achieves
to cm3, we have that the volume of the ball bearing is (3.63
REJECTION OF READINGS
- summary of notes from Ref (1) below
When is it OK to reject
measurements from your experimental results? This is a contentious
question. There are many empirical rules that have been set up to help
decide when to reject observed measurements. In the end, however, the
decision should always come down to the personal judgement of the
experimenter (1) and then only after careful consideration of the
Where an actual mistake is made
by the experimenter in taking a measurement or the measuring instrument
malfunctions and this is noticed at the time, the measurement can be
discarded. A record of the fact that the measurement was discarded and
an explanation of why it was done should be recorded by the
There may be other situations
that arise where an experimenter believes he/she has grounds to reject a
measurement. For instance, if we make 50 observations which cluster
within 1% of the mean and then we obtain a reading which lies at a
separation of 10%, we would be fairly safe in assuming that this reading
was caused by some sort of mistake on the part of the experimenter or a
failure of the measuring instrument or some other change in the
conditions of the experiment. We would be fairly safe in rejecting this
measurement from our results. (1)
"The necessity is to build up
confidence in the main set of measurements before feeling justified in
doing any rejecting." (1) Thus, there is no justification for
taking two readings and then rejecting the third because it is vastly
different to the first two. "Unless the situation is absolutely
clear cut it is by far the best to retain all the readings whether you
like them or not." (1)
We always do well to remember
that many of the greatest discoveries in Physics have taken the form of
outlying measurements. (1)
(1) Baird, D.C. (1962).
“Experimentation: An Introduction To Measurement Theory And Experiment
Design”, Prentice-Hall Inc, New Jersey