Measurement and Uncertainty
Heather Brasell
Coffee High School, 159 Trojan Way, Douglas GA 31533
(912) 384-2094 – hbrasell@coffee.k12.ga.us
How do we measure precisely?
Size, precision and units
Overhead: 2, 2.0, 2.00
Common errors
Overhead: examples
Number of digits to record
Decimal after major unit (30.3 instead of 33)
Non-decimal scales
Fractions v decimals
Zero bias
Link to plotting points on a graph
Overhead: measure and plot
Why measure?
Human senses are easily fooled
Extend the range and sensitivity of what we can measure
Why measure precisely?
Precision – tolerance - margin of error – limit of resolution
Link to significant digits and scientific notation
To determine significant differences
Overhead: error bars
Why is uncertainty important?
Limits to certainty of all human knowledge
Human desire for certainty conflicts with limits
Activities
PVC Pipe
Tree growth
Measurement activities STS
Illusion packet, rulers
PVC, PVC handout
Overhead: tree rings
Handout, graph paper
STS Handouts
specimens
Making Connections: STS – Brasell, Ely and Smith - 2/10/2003
Notes
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8
Introduction – Uncertainty in science & elsewhere
Observation – overhead – group activity
Measuring precisely – the difference between 1, 1.0, 1.00
Plotting precisely – reversing the operation
Measuring lengths – predict, measure, interpret, evaluate, conclude
Measuring length, mass, volume - predict, measure, interpret, evaluate, conclude
Growth rate of a pine tree – measure diameter and count rings (age), graph data, calculate
equation for best-fit-line (slope = growth rate)
Density of a rock – measure mass, volume by water displacement, graph data, calculate
equation for best-fit-line (slope = density)
Making Connections: STS – Brasell, Ely and Smith - 2/10/2003
Measurement and Uncertainty
All of the physical and chemical properties can be measured. Measurement involves
comparing the unknown quantity with some standard quantity.
Example: A standard kilogram is defined as the mass of a bar of platinum-iridium kept in
a controlled environment in France. Whenever we measure mass of an object, we are comparing
its mass with the mass of the standard kilogram in France. We measure how many (wholes and
parts) standard kilograms would be needed to exactly balance the unknown mass of the object.
When we say the mass is 3.4 kg, we are really saying that the object is exactly balanced by 3.4 of
the standard kilogram masses.
Measurements provide three essential pieces of information. The units tell us (a) what
size are the standard units. The number part of the measurement tells us both (b) the size and (c)
the precision of the measurement.
Example: If you measured the length of an insect in centimeters, you would say it was 5.6
cm long. You must indicate the units. Was it measured in meters, centimeters or millimeters? It
would be meaningless to say that the insect was 5.6 long. It is also nonsense to say it was 5.6
centi long. “Centi” is not a unit. It means one-hundredth of whatever unit is being used.
Precision
Precision of a measurement indicates how exactly the property has been measured. If it
was measured repeatedly, how close would the measurements be to each other? Precision
depends on the equipment being used, the techniques used, and the variability of what is being
measured.
Examples: If you had a meter-stick marked in millimeters, you could measure more
precisely than if it were only marked in centimeters. You could measure the length of an object
more precisely if it were not moving than if it were moving, if the edges are sharp than if they
were rounded or fuzzy. Your height varies by several millimeters during the day, so you couldn’t
measure your height more precisely than to the nearest millimeter.
Precision is indicated by how many digits are recorded in the measurement.
Measurements are recorded so that there is some variation allowed in the last digit only. This is
called measurement uncertainty. All measurement involves some measurement uncertainty. If
everyone in your class measured the length of a sheet of paper, the measurements would vary by
a couple of millimeters. This variation is the measurement uncertainty.
6
a
7
b
6
c
7
d
Units: m
(not to scale)
Notice there are more divisions marked on the second ruler. When you record a
measurement, you usually estimate one digit more than is given by the divisions. On the first
ruler, you would estimate one digit after the number 6 and record the measurements as: A – 6.3
m or 6.4 m, B – 7.1 m or 7.2 m. Notice there is some variation in the last digit. On the second
Making Connections: STS – Brasell, Ely and Smith - 2/10/2003
ruler, because smaller divisions are marked, you can measure with more precision. The
measurements would be: C – 6.30 m or 6.31 m, D – 7.14 m or 7.15 m.
Notice that if a measurement finishes exactly on a line, you must indicate the precision by
writing the same number of digits as if the measurement were a little before the line or a
little after the line. In the example above, the measurement for C is 6.30 m, not just 6.3
m. The final zero is indicating the precision. The ruler is marked at every one-tenth of a
meter. The measurement should be made to within one-hundredth of a meter.
In construction and manufacturing, this precision is described as tolerance – how much
variation is acceptable in the manufacturing of the object. This will be marked on blueprints.
Example: A specification may require that a part be manufactured to a length of 6.4 +/0.2 cm. The minimum acceptable length is 6.4 – 0.2 cm and the maximum length is 6.2 + 0.2 cm.
The manufacturer must produce the part so each sample is within the range of 6.2 –to 6.6 cm
long.
In manufacturing, unnecessary precision is wasteful of time and money. Imagine a piece
of wood is cut 1 centimeter too short. This is less likely to matter if the timber is used in the
framework of a house than if it is being used in a kitchen cabinet.
Example: Electronic resistors are color coded with the tolerance. If the color is silver, the
resistance must be within 1% of the rated value, but if the color is gold, the resistor must be
manufactured to be within 0.1% of the rated value.
_____|___________________|___________________|_____
2
3
4
3m
3.0 m
Units: m
(not actual scale)
between 2 m and 4 m
between 2.9 and 3.1 m
You might imagine that counting provides very precise measurements. That’s not always
true. Imagine that you are trying to count the number of people in a football stand (or insects in a
field). As you count, people are entering and leaving the stand all the time (insects are being born
and dying, entering and leaving). There is uncertainty in counting as there is in measuring.
Surveys of human preferences and opinions also have uncertainty and this is reported as
the “margin of error.”
Examples: Do you remember the 2000 presidential election? Do you remember how
difficult it was to decide exactly who people were voting for because of the hanging chads on the
ballots? The number of votes separating the two candidates was less than the uncertainty in
counting. Efforts to reform the voting system will try to reduce the measurement uncertainty, but
it can’t ever be eliminated.
During the 2000 national census, there was a heated debate about how to count people. Should
we count only the people who were included in the completed census forms? Or should
we include sample data to include people who were not counted because they were
homeless, moving, or avoiding bureaucracy? This was important because the census data
are used to decide on government funding.
Making Connections: STS – Brasell, Ely and Smith - 2/10/2003
Sometimes we want to know which of two
measurements is greater or if a property has changed. To do
this, we need to consider how much variation there is in each
measurement. Scientists often display information about the
measurement uncertainty in the tables and graphs of their
publications.
Plant growth v Treatment
86
85
84
Plant Height (cm)
Precision is very important when we are trying to
detect small differences or changes in measurements. It is not
very important when differences are comparatively large.
Example: We don’t need to measure the lengths of
pine needles very precisely to distinguish whether they come
from longleaf, slash, or spruce pine. We would need to
measure more precisely to decide which textbook used thicker
paper.
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80
79
1
2
Treatment
Example: Imagine an experiment comparing different packets of seeds of the
same species. After a couple of weeks, you found that the average heights of plants
grown from each packet were 82 cm, 85 and 84 cm. Which one grew faster?
Look at the graph on the right. The vertical lines show the amount of
measurement uncertainty for each average height. Plants in treatments 2 and 3 were both
clearly taller than plants in treatment 1. However, the “error bars” for treatments 2 and 3
overlap. We cannot determine that treatment 2 plants grew taller than treatment 3 plants.
The smallest difference that we can detect is often called the limit of resolution
of the procedure or the equipment. It depends on the detection or measurement
uncertainty. Many advances in science and technology depend on our ability to build new
detecting and measuring devices that increase resolution. They allow us to see things that
are smaller, further away, faster, slower, hotter, colder, and so on.
Examples: If you look at a bacterium under a microscope, there is a limit to how
much detail you can see. This depends partly on the magnification of the lens. Oil
immersion lenses reduce distortion effects as light travels through different materials (air
and glass). Electron microscopes use electrons rather than light. Because the electron
waves are much shorter than light waves, they can be used to see and discriminate
smaller objects. Now we have scanning tunneling microscopes that use a new technology
that allows us to see objects as small as a single atom.
With a telescope, there is a limit to what we can see. If we increase the diameter
of the mirror or lens, we can increase resolution by gathering more light to focus. Then
we can see objects that are dimmer and further away. However, if the mirrors and lenses
get too large, they bend out of shape because of temperature changes. One way to
overcome this problem is to combine several mirrors and use computers to coordinate
their focussing. Another system used in radio astronomy is to combine image data from
detectors around the world. This is called Very Long Base Array. The Hubble telescope
increases resolution another way. Because it is outside of most of Earth’s atmosphere, it
avoids the distortions caused by light travelling through air of different densities and
temperatures. These distortions are what make stars appear to twinkle.
Making Connections: STS – Brasell, Ely and Smith - 2/10/2003
3
How to Measure
1
Check the zero.
With a ruler, the zero mark is often not at the end of the ruler, which is likely to
become damaged with use. Some graduated cylinders and pipettes have two scales,
one with the zero as the top marking and the other with the zero as the lowest
marking. When you are using a balance, make sure it is pointing at zero before you
start using it.
2
Check the units being used.
Most measuring equipment will have the units written on them. However, you need to
use common sense as well. Some rulers are mislabeled “mm” when the scale is
numbered in centimeters.
3
Check the scale.
What does each division represent? If there are ten divisions between the numbers
5 and 6 on the scale, then each division represents 0.1units. If there are only five
divisions, then each division represents 0.2 units. If there are only two divisions,
then each division represents 0.5 units.
4
Record the most representative measurement.
Avoid edges and corners. If you were measuring the length of a piece of wood,
you should measure down the middle of the wood because the corners may be
rounded. If you were measuring the growth of corn plants in a field, you would
avoid the edges because they have receive less irrigation or fertilizer or more
gasoline fumes from the road along the edge of the field. The surface of liquids is
curved because of the way the molecules attract to each other (cohesion) and the
container (adhesion). This curved surface of liquid being measured is called the
meniscus. You should measure the volume of liquid by the level at the center of
the cylinder. For most liquids, this will be the bottom of the meniscus.
5
Record measurements as precisely as you can.
Making Connections: STS – Brasell, Ely and Smith - 2/10/2003
6
Use decimals, not fractions. Fractions indicate an infinitely precise number, while
decimals indicate the limit of precision.
Example: 1/3 is written as a decimal as 0.3 with a dot over the three to show that
the digit repeats infinitely. A measurement of 0.3 cm indicates a measurement
between 0.2 and 0.4 cm. A measurement of 0.33 cm shows that the measurement
is between 0.32 and 0.34 cm. This is more precise.
Making Connections: STS – Brasell, Ely and Smith - 2/10/2003