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Revised:
1/24/2011







·
Standard Units
and Measurement Systems
·
Making the (Almost)
Perfect Measurement
·
Rounding
·
Reporting
Results of Measurements
·
Summary
The
A measurement is defined as the ratio of
the magnitude (how much) of any quantity to a standard value. The standard
value is called a unit. A measurement of any kind requires both a magnitude and
a unit.
In order to begin the study of a subject,
first, we must establish a common language to be used by instructors and
students when discussing that subject. In science, an extremely important part
of this language is the measurement of various quantities, but this is just not
possible unless a good system of units has been established and is understood
by everyone. As you begin your studies, you will soon learn the definitions of
many new terms that have specific meanings when applied to your course of
study. We must develop the basic definitions necessary to take measurements and
understand those measurements.
Measurements always include error.
Part of the error in making measurements is due to the skill of the person
making the measurement, but even the most skillful among us cannot make the
perfect measurement. Regardless how small we make the divisions on our ruler we
can never be sure that the length of the object we are measuring lines up
perfectly with one of the marks. No matter how fine the measurement, there is
always another decimal places that must be estimated. Therefore, the judgment
of the person doing the measurement plays a significant role in the accuracy
and precision of the measurement.
Making accurate and precise measurements
helps us to improve our understanding of the world and enables us to predict
the outcome of future measurements. The basics of measurement must be mastered
before we can perform meaningful laboratory experiments to test the theory
leaned in the classroom. However, when using our senses to make these
measurements, we must be careful because this information can sometimes be
deceiving. For example, touching 70^{o} water outside on a very cold
day may feel warm to the touch. However, the same 70^{o} water on a
very hot day may feel cool to the touch.
Simply recording a numerical value for the
measured quantity is not sufficient to express a physical quantity. A unit must
also be indicated. For example, a measurement is taken for the length of the
laboratory table and recorded as 183. This number has no real meaning unless
expressed with a unit. A correct recording would be 183 centimeters.
There are three fundamental quantities in
which all measurements are taken:
·
Length
·
Mass
·
Time
It is very important to differentiate
between standard units and fundamental quantities. Scientists, worldwide, use
the International System of Units (the SI System) when making measurements and
calculations. The SI system is an updated version of the metric system. One
obvious difference between the two measurement systems involves units for
volume. The metric system uses liter while the SI system uses for volume. The
following table shows the fundamental quantities upon which the SI system is
based.
FUNDAMENTAL QUANTITIES 
SI UNITS (abbreviation) 
Length 
meter (m) 
Mass 
kilogram (kg) 
Time 
second (s) 
Electric current 
ampere (A) 
Temperature 
kelvin (K) 
Amount of substance 
mole (mol) 
Luminous intensity 
candela (cd) 
Almost all we know about the universe
comes from measuring these quantities. The majority of our knowledge comes from
measurements of mass, length, and time alone.
The standard units for length, mass, and
time in both the metric system and the SI system can be best remembered by
using the acronym MKS system. The three letters, MKS, stand for meter,
kilogram, and second. Chemists prefer another system, cgs, because it uses
smaller units. Here cgs stands for centimeter, gram, and second. One of the
greatest advantages of the metric system is that it is a decimal (base10)
system. Therefore, the metric system allows for easy conversion to larger and
smaller units by simply shifting the decimal.
Of course, in order to make a measurement we
first need to specify the quantity that we want to measure. It must, not only,
be measurable, but its value must also be remeasurable. Without this
"repeatability" of the measurement our results would go against one
of the most important requirements of “The Scientific Method”: "empirical
replication".
Using a measuring instrument to compare
the unknown quantity to a standard and then expressing the comparison as a
number first determines the magnitude of the quantity. A number representing
the magnitude is not sufficient to express the measurement, however; a unit
must also be assigned to the number. For example, to record a person's height
simply as 58 is meaningless. We must record it as 58 inches. All measurements,
therefore, must be recorded by listing a number and a unit.
Many instruments such as rulers, balances,
clocks, speedometers, thermometers, voltmeters, etc., are used to make
measurements. The information obtained from these instruments must be recorded
and evaluated in order to obtain a truer value of the properties of the
physical quantity. Such terms as least count, significant figures,
precision, accuracy, percent difference, and percent
error must be understood and used.
Laboratory experiments require the taking
and recording of data using a measuring instrument of one kind or another. The
fineness of the measuring scale on the instrument is known as the least count
of the instrument.
Example:
A ruler with marks 1 millimeter apart has
a least count of 1 millimeter. The smaller the least count the more precise
the instrument is said to be. For example a metric ruler that has millimeter
marks has a least count of 1 mm or 0.1 cm and when properly used it can make
more precise measurements than one with only centimeter marks.
A measurement of 1.2374 meters was made
with a stick having millimeter marks. Significant digits (or significant
figures) are digits read from the measuring instrument plus one doubtful digit
estimated by the person making the measurement. This doubtful estimate will be
a fractional part of the least count of the instrument.
Examples:
Ø 203.4
cm: least count of ruler was 1 cm, tenth of centimeter was estimated. Here, the
4 is doubtful.
Ø 4.07
cm: least count was 0.1 cm. hundredth of centimeter was estimated. Here, the 7
is doubtful.
Below are two examples of 'proper'
measurements, showing how the last digit is always ‘estimated’ or ‘doubtful’.
The doubtful digit is a 'best estimate' of the tenth fraction of the smallest
scale division on the instrument. For example, consider this measurement:
The
length of the blue line is "five point something"; more than five but
less than six. With certainty we can say it is in the range of 5 to 6. Exactly
where in that range is an estimate. We might estimate that it is 20% of the way
between 5 and 6 or 5.2.
If
we had more marks on the scale we could make a better estimate.
It
is clear that our estimate was close, but the measurement still does not align
with a mark. Also, we may have trouble reading the marks now because they are
so fine. Now we can estimate with a higher precision that the measurement is
5.28, although you might say that it is 5.27 or even 5.29.
This
method of “estimating the last digit” applies to the measurement of many
different quantities. Often, we measure a quantity indirectly. For
example, we measure the height and diameter of a cylinder, and then calculate
the volume using the measured values. Also, we may measure time using a
stopwatch. See the figure below. Here, the time is clearly between 1:00 and
2:00. The minute hand indicates the time is between 12 and 13 minutes past the
hour. As with measuring distance, we would estimate the last digit, possibly,
to be 12.4 minutes.
Note the distinction between 5.2, 5.20,
and 5.200. The three numbers indicate that the measurements were made with three
different instruments. If you do not understand why they are different,
carefully read the section on significant figures below.
When you make a measurement, it is simply
a comparison of an unknown physical quantity with a standard unit. No measurement
or answer to a problem is complete until both the number and the units have
been specified. You must always give complete answers to problems and numerical
questions in such a way that both the number and the units are clearly stated.
Also, any measurement of a real physical quantity should also be accompanied by
some estimation of the error involved in the measurement.
Errors are always present to some degree
in any measurement, and the actual accuracy of a measurement must be expressed
by the formatting of the number itself. Values read from the measuring
instrument are expressed with numbers known as significant figures. The
number of significant figures used in the number portion of any measurement
indicates how precise the data is.
To keep from implying unwarranted
precision in a number than has actually been measured or calculated, it is
often necessary to drop digits from the end of a calculated quantity. For
example, I tell you that I weigh 170 LB. If you record this as 170.00 LB, you
imply unwarranted precision. I told you my weight to the nearest pound.
However, you indicated that you know my weight to the nearest hundredth of a
pound. Similarly, attempting to measure the width of a human hair using a meter
stick with 1 cm as the smallest division is absurd. It is imperative that you
know, and can use, the rules for rounding a number to the specific number of
digits that are required in any given situation.
Below is a good example of why you cannot
simply report your answer using all digits displayed on your calculator. The
sides of the yellow box are being measured and the area is then calculated.
The least count of the ruler is 1 mm (0.1 cm). Doubtful
digit is hundredth of 1 cm (0.01 cm). 

Length is between 1.7 cm and 1.8 cm. We will estimate 1.72 cm, which is 3 SF. The doubtful digit is 2. 


Width is between 1.5 cm and 1.6 cm. We will estimate 1.58 cm, which is 3 SF. The doubtful digit is 8. 
Calculated Area = length (width) = 1.72 cm (1.58 cm) = 2.7176 cm^{2} = 2.72 cm^{2 } (to 3 S.F.) 

The true area lies somewhere in the range of: A = 1.7 cm (1.5 cm) = 2.6 cm^{2 } and A = 1.8 cm (1.6 cm) = 2.9 cm^{2} 
Once the measurements have been made and
recorded, it may be necessary to perform arithmetical operations using the
significant figures. You cannot increase the precision of measurement by
doing arithmetic operations such as addition, subtraction, multiplication, and
division. This can be confusing because the calculator will return a large
number of decimal places, especially when performing division. Go here,
for a review of Significant Figures.
Another important mathematical tool is the
use of powersof10 notation when very large or very small measurements are
recorded. Many areas of physical science have come to rely heavily on this form
of notation, and you will regularly see it in scientific writing. The exponent
on the 10 is used simply to place the decimal point; large positive exponents
represent very large numbers and large negative exponents indicate very small
numbers (less than one). Powersof10 notation is also needed to specify the
proper number of significant figures in a number when there are leading or
trailing zeros needed to properly locate the decimal point. In such cases,
powersof10 notation assures that no ambiguity in notation is present when the
number is properly written.
Your textbook also explains the use of
prefixes on units to adjust the size of these units so that they are more
appropriate in certain measurements. As an example, the prefix kilo can be
applied to a unit such as meter to construct a unit for length measurement
(kilometer) that is 1000 times larger than the basic unit (meter) and which is
more appropriate for discussing the distances between cities or the speed of an
automobile than the base unit, meter, would be.
Once you have determined how to report
your answer, that is, how many significant figures it should contain, use the
following procedures to round the number to the correct number of significant
figures.
Starting with the leftmost digit, if the
first nonsignificant digit is less than 5, drop it.
Example:
22.361 rounded to 4 significant figures is
22.36 (the 1 is the first nonsignificant digit).
Starting with the leftmost digit, if the
first nonsignificant digit is greater than 5, drop it and increase the
preceding digit by one.
Example:
15.47 rounded to 3 significant figures is
15.5 (7 is the first nonsignificant digit).
Starting with the leftmost digit, if the
first nonsignificant digit is 5, drop it and round up the preceding digit only
to make it even.
Example:
23.85 rounded to 3 significant figures is
23.8 (the 8 is already even).
Example:
23.75 rounded to 3 significant figures is 23.8
(the 7 is rounded up to make it even).
The rationale for this last rule involves
skewing your data when making a large number of measurements. Except for the
instances where you drop a 5, you will round up about half the time and round
down about half the time. Consequently, if you always round up when dropping
the 5, your data will be slightly skewed to the high side. However, using the
rule mentioned above when dropping a 5, you will round up approximately half
the time and round down approximately half the time.
Most dictionary definitions of these two
words do not clearly make the distinction as it is used in the science of
measurement.
We can never make a perfect measurement.
The best we can do is to come as close as possible within the limitations of
the measuring instruments and our ability to make the measurement.
Neither Precise Nor Accurate

This is a random
pattern, neither precise nor accurate. The darts are not clustered together
and are not near the bull's eye.



Precise, Not Accurate

This is a precise
pattern, but not accurate. The darts are clustered together but did not hit
the intended mark.



Accurate, Not Precise

This is an
accurate pattern, but not precise. The darts are not clustered, but their 'average'
position is the center of the bull's eye.



Precise and Accurate

This pattern is
both precise and accurate. The darts are tightly clustered and their average
position is the center of the bull's eye.


Attempting to reduce error in accuracy
makes for good experimental technique, but there is little we can do about
errors in precision, except using a more precise measuring device.
Experimental errors can be generally classified as being of three types:
· Illegitimate
· Systematic
· Random
ILLEGITIMATE ERROR
These type errors are due to carelessness in reading an instrument, in recording observations, mathematical calculations, or possibly an accident. Examples of illegitimate errors:
· Error in reading a scale, usually due to incorrect alignment of the line of sight.
· Recording the wrong measured value.
· Not observing the significant figures in a calculation.
· An experiment may call for a ball to be dropped and timed for how long it falls. The ball may strike another object during its fall, negating the validity of that “run”.
A measurement should never be included if it is known to be faulty. If the cart is halfway down the track before starting the clock, the measurement must be discarded. However, be careful with this. You must never take a series of measurements and choose the ones you like. All reasonable measurements must be included. This is the purpose of the lab activity – to determine if the observation agrees with the theory.
SYSTEMATIC ERROR
Systematic errors are associated with particular measurement instruments or techniques, such as an improperly calibrated instrument or bias on the part of the observer. Examples of systematic errors:
· An improperly "zeroed" instrument or an instrument that is not properly calibrated.
· Human reaction time when starting or stopping a clock. You may repeatedly stop the clock too soon or too late.
· Personal bias of an observer, who, for example, always takes a low reading of a scale division.
Avoiding systematic errors depends on the skill of the observer to detect and to prevent or correct them. Experimental physics isn't just about making measurements; it's about making meaningful measurements. Think hard about whether the number you obtain might include significant systematic errors.
RANDOM ERROR
Random
errors result from unknown and unpredictable variations in experimental
situations. Random error does not have
any consistent effects across the entire collection of measurements taken. Instead,
it causes measured values to be above and below the actual value. If the number
of measurements is sufficiently large, there will be as many values above the
actual value as there are below the actual value. Again, if the number of
measurements is sufficiently large, these random fluctuations in the
measurements would sum to zero. The effect of random errors can be reduced and
minimized by improving and refining experimental techniques and repeating the
measurement a sufficient number of times so that the erroneous readings become
statistically insignificant.
The process of taking any measurement
always involves some uncertainty. Such uncertainty is usually called
experimental error. It is expressed as a ratio of the amount of error compared
to the size of the thing being measured. For example, an error of 1 meter is
not much when we are measuring the distance from the Earth to the Moon but it
is a large error when measuring the length of a table that is two meters long.
The following two methods may be used to
calculate the amount of error.
·
Percent Error
·
Percent Difference
1. When
an accepted or standard value of the physical quantity is known, the percent
error is calculated to compare an experimental measurement with a standard.
Example:
The wellknown value of acceleration due
to gravity, to 3 SF, is . We can
employ various ways to measure this quantity. We might measure it to be
. Calculating the percent error yields:
Many texts will recommend using the
absolute value. However, if we retain the sign, we will know if our error is
above or below the accepted, or known, value.
2. Percent
difference is calculated in a comparison of two or more experimental
measurements. This is used when no standard exists, or when it is desired to
measure the precision of an experiment.
These techniques
are suitable for students just beginning to learn about errors in measurement.
However, more involved statistical techniques are employed for a more rigorous
treatment of the subject.
Error in
measurement is normal. We strive to increase the precision and accuracy of our
measurements by using a precise measuring instrument and taking great care in
estimating the doubtful digit. Also, we must be mindful of random and
systematic errors. Our ability to recognize these errors increases with experience.
However, it is fundamentally impossible to make an absolutely exact
measurement.





Any opinions, findings, and conclusions or recommendations expressed
in this material are those of the author(s) and do not necessarily reflect
the views of the 
