Error comes in two varieties, systematic and random. Measuring strikes with the wrong magnetic declination set on your Brunton would produce systematic errors, as would measuring travel times using a clock that has lost its sync with true time.
Random errors remain even after all sources of systematic error have been carefully eliminated. The best way to see random errors is to make repeated measurements of the same quantity. According to the Heisenberg uncertainty principle, even the best measurements must show Gaussian random errors at the atomic level. Usually we deal with much larger errors, and it is important to show evidence that our measurements are not dominated by systematic error.
Each student should make a table (or spreadsheet) showing all measured values versus time. If the measurements seem to change overall with time, the differences between them are likely due to be due to some systematic error (using a gravimeter or magnetometer will probably show the most systematic error). Plot a histogram showing measured values versus their frequency (how many times the measurements fell within certain ranges). If there are no systematic errors and you have more than 30 or so measurements, the random errors should force the histogram toward a Gaussian bell-curve shape.
Each student should turn in their table of measurements, and their time and histogram plots, and answer the following questions:
But all other quantities in physical calculations represent measurements, including many of the so-called physical constants. The error in all these measurements will contribute to the uncertainty of the final result. Here I show you how to incorporate the assumed Gaussian random errors of measurements into your calculations.
Given a sample of measurements with unknown error characteristics, there are other statistical methods such as jacknifing and bootstrapping that can yield the uncertainty of a computed result. The error propagation shown here is simpler and more direct, and only requires an estimate of error for each variable in an equation, instead of a measurement population.
Propagating through addition - When you add two numbers that each have error, what is the uncertainty of the sum? It is not the same as the uncertainty of just one of the addends, because the uncertainty of the other does add uncertainty. But it is also not just the sum of the uncertainties (systematic errors, though, should just be summed). If each addend can be assumed to have random Gaussian error, and the error is expressed as its standard deviation S, then we are really combining the populations to create a population of sums, which will have its own standard deviation. Thus if
a + b = cThen the standard deviation Sc of the sum is the square root of the sum of the squares of the standard deviations of a and b, Sa and Sb, respectively:
Sa2 + Sb2 = Sc2So you can see that adding two uncertain numbers increases the uncertainty, but not as much as a straight addition of their uncertainties. Note: if c is the difference between two numbers, the squares of their standard deviations still add.
Propagating through multiplication - Instead of squaring and summing the standard deviations themselves, for multiplication we square and sum the proportional errors. So if
a b = cThen we get the proportional error of the product:
(Sa/a)2 + (Sb/b)2 = (Sc/c)2So after taking proportions, squaring, summing, and taking the square root, you get a proportional error to the product c. Multiply that proportional error by c to get the actual standard deviation Sc.
As with addition and subtraction, when you divide two numbers to get a quotient, you still add the squares of their proportional errors.
Continuing calculations - Most equations have more than one addition or multiplication operation. Let's say you need to get the sum of three measurements. Then to get the error of the sum you would square all three individual errors, add them, and take the square root of the result. You would do the same with the several proportional errors of the components of several measurements multiplied together.
For a polynomial equation adding together several multiplicative terms, you find the absolute error of each term (by summing the squares of the proportional errors involved), and then adding up the squares of the absolute errors of each term.
You can derive how to handle several other types of computation. For example, if a measurement is raised to the 3rd power in an equation, that is equal to multiplying the number by itself three times. So you would take the measurement's proportional error, square it, multiply it by 3, and then take its square root to find the proportional error of the cube. Any exact geometric exponential, including a square root (multiply the proportional error by 0.5), can work that way. If the exponent is itself a measurement and has error, then you have to add in the absolute value of its proportional error without squaring it first.
More complex functions of measurements like trigonometric functions and the logarithm are much more difficult to deal with. Formally, errors should be estimated by passing the whole population of measurements through the entire computation, and then observing the statistics of the final result (histogram, mean, standard deviation, median). Jacknife and bootstrap estimates can also be used. For quick estimates, you can continue to sum the squares of proportional errors.
Writing down your values and errors - Use your resulting absolute error or standard deviation in writing down your result. For instance, in measuring a room to the nearest millimeter, I would express the final result as the average plus or minus the standard deviation:
8.72 ± 0.14 meterSelect the appropriate units or use scientific notation to make the degree of accuracy clear. I rarely ask for answers with more than three significant figures. (Exceptions include location coordinates such as latitudes and longitudes, or absolute quantities such as total gravity where we can measure to one part per million or better.) Writing the above result in millimeters, like ``8720±140 mm'' gives an incorrect appearance of accuracy.
Because standard deviations are statistical estimates, there is no point in stating errors to more than two significant figures.
From each of the four stations you have three seismograms. You can pick a P and an S arrival time from each seismogram. For each station, pick three S times independently, average them, and get the standard deviation. Do the same to get an average P time at each station, and its standard deviation. In getting the S-P times, propagate the errors through the subtractions.
Last semester in working this problem you came up with a velocity you used to translate the S-P time at a station to the distance. Assume there is 5% error in this velocity. Continue to propagate error through this multiplication. Turn in a table (or spreadsheet) showing all P and S times, averages, standard deviations, S-P times and their errors, and distances and their errors.
Draw radial distances from each station on the map as you did last semester, but now draw two circles around each station. Use the distance minus the error, and the distance plus the error. Turn in the map. Describe the error in your location. How wide is the area of intersection? Is it equant, or a lot larger in one direction than another? Make a rough estimate of the area (in km2) of the intersection.