Difference between revisions of "Error analysis"

Overview

A thorough, correct, and precise discussion of experimental errors is the heart of a superior lab report, and of life in general. This page will help you understand and clearly communicate the consequences of experimental error.

What is experimental error?

The goal of a measurement is to determine an unknown physical quantity Q. The measurement procedure you use will produce a measured value M that in general differs from Q by some amount E. Experimental error, E, is the difference between the true value Q and the value you measure M, E = Q - M.

In this context, “error” is not a synonym for “mistake,” although mistakes you make during the experiment can certainly result in errors.

Error sources

Error sources are root causes of experimental errors. Some examples of error sources are: shot noise, electromagnetic interference, and miscalibrated instruments.

Error sources can be categorized as fundamental, technical, or illegitimate. (Inherent is a synonym for fundamental.) Fundamental error sources are physical phenomenon that place a strict lower limit on experimental error. The magnitude of experimental error introduced by technical error sources can at least in theory be reduced by improving the instrument or procedure — a proposition that frequently involves spending money. Illegitimate errors are mistakes made by the experimenter that affect the results. There is no excuse for those.

Pentacene molecule imaged with atomic force microscope.^[1]

Classify error sources based on the way they affect the measurement. For example, many measurements are limited by random thermal fluctuations in the sample. In principle, it is possible to reduce thermal noise by cooling the experiment. Physicists cooled the pentacene molecules shown at right to 4°C in order to image them with an atomic force microscope. But not all measurements can be undertaken at such low temperatures. Intact biological samples do not fare particularly well at 4°C. Thus, thermal noise may be considered a technical noise source in some circumstances (pentacene) and a fundamental noise source in others (most measurements of living biological samples). There is no hard and fast rule for classifying error sources. You must think each source all the way through the system: how does the underlying physical phenomenon manifest itself in the final measurement?

Accuracy and precision

Experimenters usually worry about two types of error in measurements: random variation and systematic bias.

Error sources like thermal and technical noise introduce random variation into all measurements. Because of this, repeating a measurement twice does not give identical results. It is possible to refine your estimate of Q by averaging multiple measurements M = <M_i>. According to the central limit theorem, the uncertainty in your estimate of Q in most cases decreases in proportion to the square root of the number of measurements you average, N. Averaging multiple measurements increases the precision of a measurement at the expense of measurement bandwidth. In other words, it takes longer to make the measurement. Because the increase in precision is proportional to the square root of N, averaging multiple measurements is frequently a resource intensive way to achieve precision. You have to average one hundred measurements to get a single additional significant digit in your result. The central limit theorem is your frenemy. The theorem offers an elegant model of the benefit of averaging multiple measurements. But it is also could have been called the Inherent Law of Diminishing Returns of the Universe. Each time you repeat a measurement, the value added by your hard work diminishes.

An example of a possible systematic error source is a bathroom scale that reads five pounds too light all the time. This is called a zero-point or offset error. If you are measuring your body mass index, which is equal to your mass in kilograms divided by your height in meters squared, your result M will be smaller than the true value Q. Your result will also include random variation from other sources. Averaging multiple measurements will reduce the contribution of random errors, but the measured value of BMI will still be too low. No amount of averaging will correct the problem.

Types of errors

Systematic errors affect accuracy. Random errors effect precision.

Sample bias

Quantization error

References

1. ↑ Gross, et. al The Chemical Structure of a Molecule Resolved by Atomic Force Microscopy. Science 28 August 2009. DOI: 10.1126/science.1176210.