Difference between revisions of "Error analysis"

Overview

A thorough, correct, and precise discussion of experimental errors is the core 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.

“Experimental 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 fall into three categories: fundamental, technical, and illegitimate. (Inherent is a synonym for fundamental.) Fundamental error sources are physical phenomenon that place an absolute lower limit on experimental error. Experimental errors introduced by technical error sources can (at least in theory) be reduced by improving the instrumentation or measurement 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 into the three categories based on the way they affect the measurement. In order to come up with the correct classification, you must think each source all the way through the system: how does the underlying physical phenomenon manifest itself in the final measurement? For example, many measurements are limited by random thermal fluctuations in the sample. 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 so majestically 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 could be considered a technical source in one instance (pentacene) and a fundamental source in another (most measurements of living biological samples). There is no hard and fast rule for classifying error sources. Consider each source carefully.

Effect of error sources on measurement: random vs. systematic errors

Random and systematic errors

Imagine you are conducting an experiment that involves weighing yourself. Consider two instruments for making the measurement: an analog and a digital bathroom scale. Assume the analog scale always reads five pounds too low. Thermal noise in the amplifier of the digital scale causes the reading to vary randomly around the true value.

The analog scale introduces a systematic error in the weight measurement. The digital scale introduces a random error. The analog scale is precise. The digital scale is accurate. (The term “true” is also used.)

Because nearly all measurements involve error sources that introduce random variation, repeated measurements rarely give identical values. One way to refine a measurement that exhibits random variation is to average N measurements: M = <M_i>, i ∈ 1…N. If the measurement error is really random, the error terms E_i&nbsp= Q - M_i will sometimes be positive and other times negative; sometimes large and other times small. When multiple measurements are averaged, the errors will tend to cancel each other. Averaging several measurements increases the precision of the result at the expense of measurement bandwidth. In other words, it takes longer to make the measurement.

Some error sources result in measurement errors that do not decrease when multiple measurements are averaged. These are called systematic errors. An example of a possible systematic error source is a mass measurement using a scale that reads five pounds too light all the time. This is called a zero-point or offset error. Systematic errors reduce the accuracy of a measurement.

Bottom line: the magnitude of random errors tends to decrease with larger N; the magnitude of systematic errors does not.

The central limit theorem provides a mathematical model for averaging multiple measurements. Informally stated, the theorem says that when you add random variables, their variances add.

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 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.

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

Accuracy and precision

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

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.