PHYS 2108 Lecture Notes - Lecture 1: Error Bar, Number Sense, Significant Figures
1-1
LAB 1:
Introduction to Numerical Methods
“iee is’t siee ithout ues. We use ues to epess easueets of the old aoud
us, and then we compare those measurements to look for relationships between physical phenomena.
All of us know what numbers are, ut a of us do’t full udestad hat the mean.
Humans tend to lose track of numbers when they blow up to very large or very small values. We simply
have a hard time wrapping our heads around them. Which makes more sense: an atmospheric pressure
of 142,542 N/m2 or 1.4 atmospheres? The nearest star to the Sun is Proxima Centauri and it has a mass
of 2.447 × 1029kg—or 0.123 Sun Masses. Can you mentally picture 8,294,000 square-inches, or do you
need to hear it as 6,400 square-yards? We need context, and we get context either by using more
worthwhile units (more next week), or by comparing numbers to more familiar things.
Numbers also subtly tell us more than simply their value. For instance, if someone were to tell me a
football field (American) is 6398.74271294 square-yards, I would hear that that person is very concerned
with getting a close measurement and using extremely high-tech measurement devices; OR (more likely)
the peso does’t hae uh ue sese, is’t thikig aout the sigifiae of soe of those
digits, is lying or confused about the confidence in his or her measurement, and probably believes
everything the calculator says (to the poit he o she ight ee Ro Bugud? hiself o heself
with an obviously bogus answer).
Befoe e jup ito epeietal ok, e ill sped toda’s la deelopig soe ue sese
through various techniques in handling uncertainty and rounding, comparing values, and expressing
relationships graphically.
Confidence
There is no such thing as an exact value for a measurement (except for counting discrete objects). Any
measurement you make is limited by the precision ho lose o sall ou a measure) of the
deie used. Futheoe, the liited peisio iplies thee is soe iggle oo i hee the tue
value of a measurement lies. This uncertainty in the true value of a measurement is unsurprisingly
alled uncertainty” and it tells us something about our confidence in our measurements.
Error Bars
An error bar explicitly states the uncertainty of a measurement. For example, if the area of our
football field is 6398.4 square-feet and it has an uncertainty of measurement to 0.8 square-feet, we
express this value as 6398.4 ± 0.8 ft2. This means that there is a range where we expect the true
area to be between 6397.6—6399.2 ft2. Error bars can also be a percentage. Notice that
uncertainties are generally rounded to one significant digit and the value is rounded to the same
precision as the uncertainty.
When numbers with error bars enter into calculations, their uncertainties interact with one another
and the net uncertainty can become both complicated and very tedious. Formal propagation of
error allows the data to tell us what the uncertainty should be and is vital in scientific research.
Because this method is statistics-heavy, for purposes of this class it will be burdensome and distract
us from our course goals.
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1-2
Significant Digits and Precision
The uncertainty in a measurement is implied through its precision and number of significant digits,
that is, the number of digits that truly matter. Determining the number of significant digits in a
given value is easy.
If there is a decimal point: come from the left, ignoring any zeroes until you hit something different.
Cout the digits eaiig iludig a zeoes sadihed etee othe digits.
Four significant digits
Precision implied to be to nine decimal places
If there is no decimal: come from the right.
Two significant digits
Precision implied to be to the tens place
SPECIAL CASE: sometimes zeroes are significant. If we add , we will find
out that the sum is significant to the tens plae. We eithe state outight to two
sigifiat digits o use a oea as a eide to out oe of the zeoes as i
.
We have a set of rules for significant digits when they enter into calculations. If a footall field’s
length (L) and width (W) are measured (with fractions-of-inches precision) to be 119.953123 yd and
53.34371 yd, find the perimeter and area of the field.
Addition/Subtraction Rule: The final result should be to the same precision as the least precise
measurement. Ex: If the perimeter P = L + W + L + W…
53.343710 yd
119.953123 yd
53.343710 yd
+ 119.953123 yd
346.593666 yd
345.593670 yd
Sum values keeping all digits and precision
until the end.
53.34371 contains the fewest decimal places:
five.
Round the final result to five decimal places.
Multiplication/Division Rule: The final result should be to the same number of significant digits as
the measurement with the fewest significant digits. Ex: If the area is A = L × W…
119.953123 yd2
× 53.34371 yd2
6398.74460690633 yd2
6398.74460690633 yd2
6398.74560690633 yd2
Sum values keeping all digits and precision
until the end (or to the limits of your
calculator).
53.34371 contains the fewest significant
digits: seven.
Round the final result to seven significant
digits.
B shoig fiftee deial plaes, ou’e iplig uh highe peisio ad uh oe
confidence in your answer than is warranted. Significant digit treatments help keep you from
unintentionally lying to your reader or yourself.
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find more resources at oneclass.com
1-3
Tolerance
Even assuming you used high precision instruments and 6398.745 yd2 is an honest and factual
easue of the footall field’s area, I still do’t u it. I know a few things about football fields. The
borders are several inches wide, and the edges of those borders are jagged from being painted on
grass. I know that there is enough lab error in this measurement (by no fault of the measurer) that
those extra digits are comically useless. For me, the field is 6400 yd2.
Tolerance eas ho uh impeisio ill ou toleate? What is good eough? This is often
eithe a judget all ou, o a aita deisio ased o ou eeds. Fo soe of ou la
apparatus, the tolerance on the size of the parts from the machine shop needs not to be extremely
precise. It just needs to be tight enough that the stuff is’t so loose it falls apart. Aircraft parts need
to be much more exact. Setting tolerances sounds like the easiest method for uncertainty listed
because it can be whatever you want, but determining toleaes fo ou gut relies on
experience. It is more art than science, and doing it well by instinct is the hardest part of developing
ue sese.
Comparing Numbers
A big part of analyzing data is comparing values: either in measuring two separate things, comparing
before-and-after an event, or in comparing measurements to predictions. We have several tools.
Overlapping Error Bars
We never know a true value, but we know a range that probably contains it. Ideally, if the error bar
from one measurement contains the value of
another, we can fairly confidently say the two
easued alues ae the sae. Fo eaple,
100cm ± 15cm and 120cm ± 30cm are the same
because the range of 90cm – 150cm contains
the value 100cm.
If the alues ee pehaps ± ad ± , e ould still sa the ae the sae.
Ee though eithe age otais the othe’s epoted alue, the uetait epesets a
possiilit that the tue alue lies
anywhere along it. The fact that there
is overlap between the two error bars
means that the two values could be the
same. We may have less confidence the values measured were the same than before, but we still
have some confidence.
When the two measurements have no overlap whatsoever, 100cm ± 15cm and 150cm ± 30cm, we
can safely say they are of different values.
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Document Summary
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