The Role of Measurement Error

For a lot of students, as well as the general public, there is a wholesale failure to understand and apply the concept that every measurement has an error.

There is simply no such thing as a perfect measurement or a perfect detector. All dectetors/measurements have random noise associated with them.

In the case of measuring the positions of planets with your naked eye, there is a lot of random noise and lack of precision and hence, within the precision of the measurements, the Ptolemaic model worked okay to predict the positions of the planets. Therefore it seemd to work. If anyone would have developed an instrument to measure planetary positions more accurately (e.g. using a telescope), the Ptolemaic model would have not correctly predicted the positions of the planets.

The effect of random noise is that no two measurements are ever exactly the same. Now if the noise is sufficiently small, the average person will not notice this effect in there every day life. Without such notice, the individual labors under the illusion that measurements/science are perfect Nothing could be further from the truth.

To begin with, we will apply the concepts of errors to something that you care about - your exam scores!

On any exam, your score reflects two things:

• Your ability in applying the material learned in this course
• Errors associated with how the exam was graded. These errors come in two forms; one is much worse than the other:

  1. Random error (can be corrected for - see below)
  2. Systematic Error (extremely serious if you don't know it exists)

  These Errors works like the following:

  

  What we measure is X but what we are interested in is the distribution of the true variable, T. To measure T, however, we have to know what the random error, e_r and systematic error e_s is.

  Without knowledge of e_r and e_s , T can never be accurately measured. This potentially is a huge problem.

  What is e_r

  

  Random errors increase the variability around the mean (in fact this process in nature is what drives genetic evolution).

  Random errors are associated with apparatus or method used in obtaining the data. All data sampling is subject to random error, period. There is no way to avoid it. You will definitely be dealing with random error and noise in this class.

  Note that in most detectors random error is a fixed quantity so that the accuracy of the measurement depends upon the amplitude of the measured phenomena. This idea will become more apparent to you later in the term when we simulate the detection of extra solar planets, among other things.

  For now, here is a simple example.

  Suppose I have a thermometer with a fixed random error of +/- 2 degrees. That is the accuracy of this device. Well, if its really 100 degrees out side, then this device will measure the tempearture to an accuracy of 2% (2 out of 100). However, if its 20 degrees outside, the device will only measure the temperature to an accuracy of 10% (2 out of 20).

  What is e_s ? (Systematic Error; often called calibration error).

  

  Systematic errors mean that different methods of measurement are being applied to the same variable. This means that the position of the mean is strongly effect. For example, suppose there are two patrolment on the freeway both with identical radar guns. Except that one of them systematically reads 5 mph higher than the other due to a "calibration" error back at the station. Which policeman do you want to speed by?

  The role of BIAS in astronomical measurement is huge and its another topic you will be exposed to throughout the term.

Now we will run a laptop simulation to get a feel for random and systematic errors.

Click here to run the simulation

Procedure:

• All measurements will be made at 60 mph
• Pay attention to what officer your asked to be
• When you have made your measurement, then "publish it to global view"