On the first exam, your score reflects two things:
1. Random error (can be corrected for - see below)
2. Systematic Error (extremely serious if you don't know it exists)
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, er and systematic error es is.
Without knowledge of er and es , T can never be accurately measured. This potentially is a huge problem.
What is er
Random errors increase the dispersion. These 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. However, if you know what the value of er is, you can "subtract" it out of the data.
Here is how you due to subtraction. In a bell-shaped distribution, the various components are added or subtracted in quadrature. What does that mean?:
(true dispersion)2 = (measured dispersion)2 - er2
Example: Suppose a measured distribution has a dispersion of 8 units and we know that the measuring error er is 6 units. What is the true dispersion?
What is es ?
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?
Your exam was graded by 4 different people although all have the same exam key. The exams were randomly sorted as they were handed in and the 4 graders randomly picked a pile. Each pile has 30 exams.
Since the intrinsic distribution of exam scores will be normal (bell- shaped) then by the sampling principles established earlier we know that 30 samples is enough to accurately reflect the distribution.
Therefore, each of the 4 piles should have the same mean and dispersion. Differences in grading style, however, will cause each grader to have a slightly different mean and dispsersion.
es will then manifest itself by different mean scors for different graders. This can be corrected for easily.
er manifests itself by different dispersions for different graders. This is due to random errors associated with grading the questions and assigning a point value in a lightly subjective manner.
For this exam, your dispersion score was calculated by the instructor after er and es had been determined. In principle, this needs to be done for every exam you take in large classes. Most instructors don't do this, period (its too much work).
For this exam:
Introduction to Measurement Errors:
In class exercise.
On a piece of paper, write down your estimates for the following
quantities and give them to one of the TAs.
On a piece of paper, write down your estimates for the following quantities and give them to one of the TAs.
Understanding the role of measurement errors is crucial to proper data interpretation. For instance, the measured dispersion in some distribution represents the convolution of
In general, you only care about the intrinsic dispersion in some distribution. That is, you don't want to have the dispersion dominated by measurement error or poor precision because then you can't draw any valid conclusion.
Example: Column 1 contains the data that was measured with good precision. That is, the measuring error of the instrument was less than 0.1. Column 2 represents the same data that was measured with and instrument that had a measuring error of +/- 1 unit:
Clearly the first column is a better measure of the intrinsic distribution of the sample than the second column. Essentially the numbers in the second column are meaningless.
In principle, every measurement has an error associated with it and hence a measurement is only as good as its error. Knowing the size of measuring or sampling errors is often difficult.
For some kind of sampling, error estimation is straight forward. For instance, opinion poll sampling has an error that depends only on the Number of people in the sample. This error has to do with counting statistics and is expressed as
Conclusion: Always ask what the measuring errors are!!!
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