          ## Experimental Error

Error (or uncertainty) is defined as the difference between a measured or estimated value for a quantity and its true value, and is inherent in all measurements. Knowledge of the type and degree of error likely to be present is essential if data are to be used wisely, whether the data being considered were measured personally or merely read from manufacturer's data sheets for a material or component. In medical research, biology, and the social sciences, the plan for the data acquisition and analysis is the heart of the experiment. Engineers also need to be careful; although some engineering measurements have been made with fantastic accuracy (e.g., the speed of light is 299,792,458 1 m/sec.), for most an error of less than 1 percent is considered good, and for a few one must use advanced experimental design and analysis techniques to get any useful data at all. Making measurements and analyzing them is a key part of the engineering process, from the initial characterization of materials and components needed for a design, to testing of prototypes, to quality control during manufacture, to operation and maintenance of the final product.

Reported experimental results should always include a realistic estimate of their error, either explicitly, as plus/minus an error value, or implicitly, using the appropriate number of significant figures. Furthermore, you need to include the reasoning and calculations that went into your error estimate, if it is to be plausible to others. An explicit estimate of the error may be given either as a measurement plus/minus an absolute error, in the units of the measurement; or as a fractional or relative error, expressed as plus/minus a fraction or percentage of the measurement. The advantage of the fractional error format is that it gives an idea of the relative importance of the error. A 10-gram error is a tiny 0.0125% of the weight of an 80-kg man, but is 33.3% of the weight of a 30-g mouse.

Errors may be divided roughly into two categories: Systematic error in a measurement is a consistent and repeatable bias or offset from the true value. This is typically the result of miscalibration of the test equipment, or problems with the experimental procedure. On the other hand, variations between successive measurements made under apparently identical experimental conditions are called random errors. Random variations can occur in either the physical quantity being measured, the measurement process, or both. We will outline statistical procedures for handling this type of error.

In reporting experimental results, a distinction should be made between "accuracy" and "precision." Accuracy is a measure of how close the measured value is to the true value. A highly accurate measurement has a very small error associated with it. Note that in experimental work the true value is often not known, and thus what is reported must be an estimated accuracy or error. Precision is a measure of the repeatability and resolution of a measurement -- the smallest change in the measured quantity that can be detected reliably. Highly precise experimental equipment can consistently measure very small differences in a physical quantity. Note that a highly precise measurement may, nevertheless, be quite inaccurate. High precision in a measurement is a necessary but insufficient condition for high accuracy.

EXAMPLE: a typical 3 1/2 digit voltmeter can measure voltages between -199.9 volts and 199.9 volts, when set to its 200 volt range. Suppose that in measuring a voltage an engineer obtains an average reading of 47.1 volts, and that the right-most digit flickers up and down between 0, 1, and 2 in an apparently random way. Thus, the precision is approximately volts in absolute terms. The fractional or percentage precision is volts/ 47.1 volts, or . The precision can be estimated from the measurements obtained; the accuracy must be found by comparison to an accepted voltage standard. Looking at the manufacturer's specification sheet, which includes the results of such a comparison, the engineer finds that the rated accuracy of the voltmeter is . Thus, the accuracy is an order of magnitude worse than the precision. (The accuracy of a voltmeter is only as good as its resistors and other components. Resistor values drift as the resistors get older, and they vary with temperature.) In terms of absolute error, the measurement should be reported as 47.1 (0.2)(47.1) volts, or 47.1 1 volts (not 0.1 volts, the estimated precision).

 Introduction Main Body •Experimental Error •Minimizing Systematic Error •Minimizing Random Error •Propagation of Error •Significant Figures Questions