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Return to Dream Catchers start
page.
Revised: 9/15/2008
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·
Standard Units
and Measurement Systems
·
Making the
(Almost) Perfect Measurement
·
Rounding
·
Reporting
Results of Measurements
·
Summary
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 70o water outside
on a very cold day may feel warm to the touch. However, the same 70o
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 (base-10) 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 re-measurable. 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
guess.
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 don't
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 perfect 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.
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The least count of the ruler is 1 mm (0.1 cm). Doubtful
digit is hundredth of 1 cm (0.01 cm). |
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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. |
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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. |
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Calculated Area = length (width) = 1.72 cm (1.58 cm) = 2.7176 cm2 = 2.72 cm2 (to 3 S.F.) |
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The true area lies somewhere in the range of: A = 1.7 cm (1.5 cm) = 2.6 cm2 and A = 1.8 cm (1.6 cm) = 2.9 cm2 |
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 powers-of-10 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). Powers-of-10 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, powers-of-10 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 left-most digit, if the first non-significant digit is less than 5,
drop it.
Example:
22.361
rounded to 4 significant figures is 22.36 (the 1 is the first non-significant
digit).
Starting
with the left-most digit, if the first non-significant 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 non-significant
digit).
Starting
with the left-most digit, if the first non-significant 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 t