Estimating Error

Types of Errors

A common practice in any science is measurement. Measuring is always associated with error. The error arises for various reasons. We discussed in the last section that the precision of an instrument is one reason that error must be present in measurements. Other reasons that error exists are:

Systematic errors. These errors are ones which don't cancel one another upon repeated measuring. A simple source would be not zeroing or taring a scale. Another example is using an ultrasonic ranger - a device which sends pulses of sound and waits to hear an echo in order to determine distances - without accounting for air temperature properly. The speed of sound depends on temperature, so if the device is set up to assume a temperature of 22 C and the room happens to be much warmer or cooler there will be a systematic error in all the measurements. The good thing is that for each five degree Celsius deviation from expected temperature the ranger would only incur a 1% error. We will expect most times that our equipment is calibrated such that these are not significant. Nonetheless, keep this option in the back of your mind if your lab data looks bad and you are running out of ideas about why your results are so bad. Calibration should fix systematic errors most times.
Random errors. These are ones which due to their random nature will cancel one another after repeated measurements are taken. A bad dart thrower will sometimes be too high and sometimes too low such that after many throws the average height will be at the bulls eye even if no individual dart ever was. All instruments have random fluctuations. The fact that these cancel upon repeated measurement is a good thing if you wish to use a low precision device to make a precise measurement. We'll discuss this shortly.
Blunders or mistakes. Examples of this type of error are things like recording an incorrect value from an instrument, or perhaps using the instrument incorrectly. These should stick out from the other data points and should simply be discarded from the data set rather than included in analysis of the data. Sometimes discretion must be used. For instance, in hand timing with a stopwatch the person doing the timing might have the "feeling" that they stopped too early. It's a good idea to remove such data points from the set. There is no reason to muddy a decent data set with points of questionable quality.

Estimating Random Errors

Random errors are the ones we will mostly handle. In other words, we will do our best to avoid blunders, and will expect that in almost all cases our instruments are well-calibrated. We will often attempt to quantify a value of some physical constant like the acceleration of gravity or the stiffness of a spring. While such values should be the same every time we measure them (within reason), due to the precision of our instruments we will likely get different values every time we measure them. For instance, next week in lab we will measure the acceleration of a falling object due to gravity. In units of meters per second squared, the value near sea level is supposed to be 9.81. If you attempt to measure it three times, you may find values like: 9.68, 9.77, 10.03. What are we to do with those numbers? The idea is that we want to report not the value we can look up in a textbook, but rather the value, along with its associated error, that our instrument measured. We will do this by using some statistics which is built into GeoGebra, but which you have possibly used before in other science classes, or perhaps have heard of in a statistics class. We need to be clear on a few terms like: mean, standard deviation, standard error, confidence interval, normal distribution. I will briefly describe these below, but if my terse explanations don't do it for you, please google the terms.

Random Errors and Statistics

As I'm sure you know, the mean is just another word for the average value. I won't spend any time on that one. Random errors tend to produce measured values that center around the mean like a bell curve, which to engineers, mathematicians and scientists is called a normal distribution or a Gaussian distribution. Commit these terms to memory since you plan on being an engineer or scientist. Better yet, use one or the other of those terms in place of "bell curve" from this day forward... unless you're speaking to a non-technical audience. The peak of the normal distribution is the mean because the distribution has symmetric tails. There are many other types of distributions that are not symmetric like this. You can play around with them in the probability view in GeoGebra. For a Gaussian or normal distribution, the width of the distribution is narrower the more precise your instrument. That width can be quantified by the standard deviation. When the standard deviation is large, the "bell" is wide, when it's small the "bell" is very narrow.

Understanding Random Error

I will walk you through using GeoGebra to understand random error. And how a larger number of measurements allows a better prediction of the mean.

The sequence command allows repeated operations. We will make N points, where N is an integer. First define N by typing N=1 in the input cell. Allow a slider to be made.

Right click the slider and make the minimum 1, the max 50, and the step 1.
Next type sequence[(random()-0.5,random()-0.5),i,1,N] The random() command makes a random number between 0 and 1. So when we subtract 0.5, it makes the random number between -0.5 and 0.5 such that the center is the origin. Putting the two random statements in parentheses defines the x and y coordinates of a point. After the commas we iterate a variable named 'i' starting at value 1 and ending at N. In other words, repeat this process N times. The 'i' is not really being used here in the definition of the points (it could have been), but is just a necessary part of the syntax.

To visualize the mean of the N points, type (mean(x(list1)),mean(y(list1))).This creates a single point with x component being the mean of the x components of the list of points we created, and the y component being the mean of the y components of the points.
Each time you hit control-r (for refresh) on the desktop version of GeoGebra (not here), a new set of random points will be created and the new associated mean will be shown. Notice that when N is small, the mean jumps all over the place, but when N is large, the mean stays near the origin. Mathematically, the mean stays closer by a factor of
What this implies is that any instrument with an associated random error will be able to give you a precise value for some quantity so long as you are willing to make repeated measurements of the same quantity over and over again! Measuring 10 times reduces the error by times.

Standard Deviation

The standard deviation which describes the spread of a distribution is the root mean square (RMS) deviation of a data set from its arithmetic mean. The root mean square just means that we take the square root of the mean value of the sum of squared deviations of the points from the mean value of the points. It does a good job of representing the "spread" of the data points about the mean. The standard deviation is denoted with a Greek lower case sigma (like our 's'), and is calculated using the formula

where

represent the individual measurements, and N is the number of measurements or data points. While I don't put much stock in the measurement of IQ - even though mine is a bit excessive;) - it is a good example of standard deviation. If everyone on earth was equally intelligent, everyone's IQ would be 100 since that is the mean value of test takers. Each 15 points above or below represents one standard deviation above or below average. One fact about a normal distribution like this is that we can quantify population percentages within certain ranges. For instance, 68.3% of a population always falls within one standard deviation from the mean. 95.4% falls within two standard deviations from the mean, and 99.7% falls within three standard deviations. In the present context this means that 68.3% of the population has an IQ that falls between 85 and 115, and that 95.4% of the population falls between 70 and 130. If the population IQ standard deviation were 5, we wouldn't find many really smart or really dumb people. If it were zero we'd all be equally smart on IQ tests. If it were 50 many people would be levitating things with their minds and many others would be dragging their knuckles down the sidewalks while drooling on the way to work. A point to realize, however, is that people within populations vary in their IQs. Thus we expect a spread. More measurements of more people's IQs would not change the standard deviation in any significant way. This is in stark contrast with standard error in which we measure a single quantity multiple times rather than many different quantities (IQ scores of individuals) once. In lab we will often measure the SAME thing multiple times with hopes of determining it's real value. Recall that we measure multiple times to reduce the associated error of the measurement. Doing this is more like having a population of identical people and having a poorly written IQ test or a problem with the grading machine that sometimes awards points for wrong answers and sometimes counts right ones wrong. So even though everybody really has the same IQ of 100, one test comes back 83, the next 117, the next 101, etc. If we only hand out 2 or 3 tests to this population of identical people, we will have only a very rough idea of the population average IQ, but if we hand out many tests, we will know it well. In this sense the uncertainty, or error in our estimation of the IQ should get smaller and smaller the more tests we administer or the more measurements we take. In our physics lab next week, the more times we attempt to measure gravitational acceleration, the more precisely we can report it. To report it, we use the standard error, which accounts for the decrease in uncertainty with an increase in the number of measurements.

. What we do with the standard error is use it to write the uncertainty. Recall that 95.4% of a population falls within 2 standard deviations from the mean. If we want exactly 95% accounted for, it will be 1.96 (not 2) standard deviations below and above the mean. So if we write our value - for instance that of gravitational acceleration with error margin included, we will write it such that we are 95% confident that it falls within a value of

. Use of this 95% confidence interval is common in the scientific literature. That is what we will use in lab. In some areas such as particle physics, researchers demand a much higher confidence level. GeoGebra will do this math for you in the probability tab once you know the number of data points, the mean of the data and the standard deviation. We will do some of these exercises in class.

Significant Figures and Standard Error

There is sometimes a discrepancy between the usual rules of significant figures, and the rules we should employ when reporting standard error of measurements. Given measurements of what is supposed to be gravitational acceleration, the mean is typically supposed to be reported to the number of significant figures that each individual measurement has. The problem is that the error might be too small such that it is zero to that number of digits. For instance, what if reported to three sig figs we got a value of

What we must do in such cases is report the first non-zero digit of error. So we must instead report

(assuming the next digit was '3' and the error is known to be 0.004 when containing a single non-zero digit) so that the error is shown in the result. This is a more honest way to report the value rather than truncating the digits and reporting no error, since no error implies an infinitely precise measurement. Please follow this convention in your labs. Also, keep in mind that this wouldn't be a concern if the data had a larger associated error such that the result was

. In all cases, the precision of the measurement and error must match.

Calculating Standard Error

Imagine we have a data set with N=5 values. The mean of the values is 10.35 and the standard deviation is 0.11. Within the probability view as shown below we can find how to write the measured value along with its standard error.
We have to choose the statistics tab and select Z Estimate of a Mean from the drop down menu.
Then just enter the known values of N, and the mean while using 0.95 (95%) for the confidence level.
Scroll down to see the reported value with associated error. Note that it is not likely to be given with the appropriate number of significant digits!
With these values we would write it as , where we rounded the error up. It is possible that if we had lower precision data with only three significan figures that we would need to write this as .