Astro 8A: Observational Astronomy
Good Lab Science Practices
Sources
of Error
Measurements can't be done to infinite accuracy.
There are lots of reasons why the accuracy is limited.
Random errors. A random error is one which, if the measurement is repeated
over and over, will average out to about zero. In other words, the measurement
has no particular bias towards being too high or too low relative to the true
value.
1. Instrumental Precision. Your measuring instrument
will have limited precision. An example, measuring your
height using a meter stick. You can only read the meter stick to maybe
a millimeter with your eyes. If you use a laser interferometer, you could
increase your precision down to millionths of a millimeter. If you were judging
by eyeball the brightness of a star, your precision will only be maybe 0.2
or 0.3 magnitudes. But if done honestly and objectively, without a preconception
of what the brightness should be, each student's measurement is an independent
random variable and will show no overall bias.
2. Blunders. These are perhaps the first
thing that you think of when you hear the word "error". You screwed
up. You made a big mistake. Those are called "blunders" in the science
business. If these are small, you may not notice them. If they're big, you
should be able to spot them by applying some common sense. These should be
controllable if you are careful!
Systematic
Errors. These are the ones you really
want to minimize. A systematic error is biased towards giving an answer which
is consistently too high or too low, and thus they do not average out to zero.
1. Cheating by Copying someone else. This is the one I most worry
about with beginners. If 20 people measure a star's brightness by copying
the one student who seems the "smartest", yet his measurement is
off by 0.4 magnitudes, all of their magnitudes will be off 0.4 magnitudes
and all in the same direction - no amount of averaging will help!. Even if
students add in a small difference to cover up the copying, those difference
will be based around a single measurement with its inherent difference from
the true value, and thus the results will average around a biased answer.
2. Faulty instrument. Maybe your meter stick wasn't
made well and is just too short. Every measurement you make
with that stick will show a systematic error; the measurments
will be too low.
To say more, you'll generally have to know the details of the situation to
figure out what other systematic errors might be lurking.
Significant
vs. Insignificant Digits
Most calculators will display a ton of digits.
10, 15, maybe even 20 digits. Often, students will
do a calculation and write down every digit on the calculator. Don't you be
one of them! Here's the deal - when you have input values for a calculation,
those numbers have a limited accuracy, as described above. A little common
sense will tell you what accuracy you can make a measurement, assuming systematic
errors are not significant. If you're unsure, you can discover it with a tiny
experiment (science!). Do your measurement, then
make a conscious effort to forget the answer you got, and make another measurement.
Repeat. See how your answers differ. Example; you measure an angle on a lab
sheet with an ordinary protractor, getting 55.5 degrees. You repeat, and get
54.6 degrees. Repeat, and get 54.9 degrees. Looks like you your final answer
should be 55.0 degrees - the average of the three. And your accuracy is about
+-0.4 degrees. So the first two digits, 55, are pretty solid,
and the 3rd digit is kinda flakey, the .0.
It's best to report 1 digit more than your last digit of solid confidence.
You've got 2 or maybe 2 and a half significant digits,
so you should report 3 digits. You can even write down 4 digits, no complaints
from me. But if you write down every digit from your calculator, you'll lose
some grade points. Now notice also that I wrote down 55.0 degrees as my final
answer. Sometimes students will write down 55 and not the .0. Big
mistake! If I see a rounded-off number, my impression is "Hmmm.
55, not even a decimal point. Guess the error must be up to a few degrees".
Whereas if you report 55.0 degrees, my response will be "Hmmm. 55.0;
the error must be only a few tenths of a degree". In other words, you're
communicating the accuracy of your measurement by how many digits you write
down.
So how many digits should you report? I
can tell you what the accuracy is of the numbers which I include on your labs, but only
you can estimate how accurate are your own numbers. Again, you can discover for
yourself by re-doing the measurement a few times and seeing how they scatter.
Dimensions
and Units
In the "real world", measurements
are usually dimensional: They are measurements of length, or time, or brightness,
or mass, or temperature, etc. The fundamental dimensions of physical quantities
are length, mass, and time. But many useful quantities have dimensions which
are combinations of these. Energy, volume, density, luminosity, temperature...
are all in this category, sometimes abbreviated with their own names.
A given dimension is expressed in units of
measure, or units for short. There are a variety of units, convenient in different
contexts. Your height may be expressed as 5' 8", or 68 inches, or 1727.2
millimeters, or 0.001073 miles. They're all the same physical length, just
expressed in different units.
The nice thing about units is that they obey
the same math - they divide out, they square root, etc. - just like regular
numbers. So for example 10.2 meters
per second squared times 2 meters squared per degree Kelvin would be
(10.2
m/sec2) x (2.0 m2/K) = 20.4 m3/(K sec2)
and...
= 4kg/m
As in all cases, you must do your own work. However, I will help
you during your actual labs with relevant examples up on the whiteboard
in front of the classroom. When you
express answers to your labs, if they are dimensional then in order to get
full credit you must include the units! Ask me if you need any clarification.