Error Propagation

Errors Most of what we know is derived by making measurements. However, it is never possible to measure anything exactly (eventually Heisenberg's uncertainty principle would bite you!). So when you measure something, you have to say how precisely you have measured it.

The idea of error A measurement may be made of a quantity which has an accepted value (such as c). However, this value itself is just measured and has an associated error. They are just errors measured by other people. You cannot translate your value to theirs.

The idea of error Errors are also not 'blunders' or experimental mishaps which can be fixed. If a telescope is bumped during an exposure resulting in two sets of stars, the image is simply deleted and redone. If it's not caught at the time, then this datum is discarded.

The idea of error Errors occur for all measurements. It is impossible to know exactly how far off a measurement is. Otherwise, the measurements would just be 'adjusted' to the proper value. But this is not possible.

The classification of error Errors occur in two types: systematic and random.

The classification of error Systematic errors are errors which shift all measurements in the same way. Things like improper equipment calibration, thicker atmosphere than anticipated, etc. An example would be switching from instrumental magnitudes to a 'standard' system. Our instrument shifts all our measurements one way and so we work out a correction.

The classification of error Random errors displace measurements in an arbitrary direction (they scatter measurements about a value). They cannot be eliminated, but usually can be reduced (by making multiple measurements). Each measurement of a star includes random fluctuations caused by seeing, CCD noise, random sky noise, etc.

The classification of error Random errors displace measurements in an arbitrary direction (they scatter measurements about a value). They cannot be eliminated, but usually can be reduced (by making multiple measurements). Example: If a particular isotope has a decay rate of 1,000 decays per minute, then a 5 minute measurement will count, on average, 5,000 decays. However, very few individual measurements would be 5,000 exactly.

The classification of error Random errors: One way to help is to make many measurements. Example: If a particular isotope has a decay rate of 1,000 decays per minute, then a 5 minute measurement will count, on average, 5,000 decays. The first measurement may have been 4,723. If this is the only measurement, then we would produce an incorrect decay rate.

The classification of error Mean (average) value.

The classification of error Probability of obtaining result x: Where is the most probable value and  is the standard deviation, which is the width of the distribution. This is a Gaussian distribution and for large numbers, this usually describes the distribution of values.

The classification of error Probability of obtaining result x: The formula on the left is the standard deviation, which is the error and on the right is the true standard deviation, which is also the error. The right formula is because we are not using the true average value, but a compilation of values to produce an average.

The classification of error Standard deviation. The standard deviation means that if you take a measurement, 68% of the time, it will be within one standard deviation (1  ) of the average. Of course this means that 32% of the measurements are outside of 1 .

The classification of error Standard deviation. If you make some measurements and determine that  =25. Then 2  =50, 3s=75, and n  =n*25. However, the meaning is different. 1  contains 68% of the measurements, but 2  contains 95% of the measurements, 3  contains 99.7% of the measurements, and 4  contains % of the measurements.

The classification of error Confidence levels: These can sometimes be called confidence levels. So if someone says they have a 95% confidence level, that means 2 . But it could also be a different type of test (like false-alarm probability) too!

Error propagation: If you add or subtract measurements: If you multiply or divide measurements:

Weighted average: Where w i are the 'weights' for each measurement. It is common for w=1/  2. Then the weighted standard deviation is:

Least squares minimization (includes linear regression) When y is the smallest, then the line best represents the data. You may have used this in a spreadsheet with the command slope or linest.