### 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. Random errors can result from a number of causes, such as...

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. Averaged together, they should come close to the real measurement.

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. Here's the most relevant ones I can think of for our class...

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.