Respected Professor Kihyeon Cho EXPERIMENTAL ERROR Course Title: Experimental Method and Data Process Respected Professor Kihyeon Cho Prepared By: A.K.M. Moinul Haque Meaze Student ID: 2003419008 Department of Physics Kyungpook National University Republic of Korea meaze@hep.knu.ac.kr 18th May 2004

To Err Is Human, To Describe The Error Is Sublime -Cliff Swartz, Physics Today,37(1999) 388 BASIC CONCEPT 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. meaze@hep.knu.ac.kr 18th May 2004

How, then, can we know the true value of a physical quantity? No physical quantity can be measured with perfect certainty; there are always errors in any measurement. This means that if we measure some quantity and, then, repeat the measurement, we will almost certainly measure a different value the second time. How, then, can we know the true value of a physical quantity? The short answer is that we can't. However, as we take greater care in our measurements and apply ever more refined experimental methods, we can reduce the errors and, thereby, gain greater confidence that our measurements approximate ever more closely the true value. meaze@hep.knu.ac.kr 18th May 2004

PURPOSES OF ERROR STUDY Understand how to measure experimental error 2) Understand the types and sources of experimental errors 3) Clearly and correctly report measurements and the uncertainties in those measurements, and 4) Design experimental methods and techniques and improve our measurement skills to reduce experimental errors. meaze@hep.knu.ac.kr 18th May 2004

TYPES Human error (a mistake) occurs when the experimenter, make a mistake.  Examples would be when we set up our experiment incorrectly, when we misread an instrument, or when we make a mistake in a calculation.  Human errors are not a source of experimental error; rather, they are “experimenter's” error.  Do not quote human error as a source of experimental error ! meaze@hep.knu.ac.kr 18th May 2004

An absolute error is an error expressed in physical units An absolute error is an error expressed in physical units. For example, if we measure the acceleration due to gravity in the lab room to be 9.7 m s^-2  , the absolute error is -0.1 m s^-2  . Absolute errors should always have the physical units indicated. A relative error, or fractional error, is an error expressed as a fraction of the value measured or the true value (if the error is small, it makes little difference). In the above example, the error was 0.1/9.8 = 1% relative to the true value. If the true value is not known, relative errors are given with respect to the measurement. Relative errors should always be displayed as a percentage, to avoid confusion with absolute errors. meaze@hep.knu.ac.kr 18th May 2004

An uncertainty is a range, estimated by the experimenter, that is likely to contain the true value of whatever is being measured. For example, if we measure a distance with a meter stick we usually assign an uncertainty of ± 1mm to the result. Uncertainties can be expressed in absolute terms or relative terms, just as errors can. A confidence level is the probability that the true value of our experiment falls within a given range of uncertainty. Confidence levels can be exactly defined we have a good understanding of the nature of our errors. meaze@hep.knu.ac.kr 18th May 2004

An illegitimate error is not one born out of wedlock, but a one type mistake in the procedure which produces a bizarre value. If we know we made a mistake (for example, kicking the equipment in frustration) then we can just throw out that measurement. Usually the mistake is more subtle, for example misreading a display or an unexpected power surge in the equipment. In this case, people usually use some statistical criteria to throw out data which are well outside the normal range of possibility. The precision of a measurement is the total amount of random error present. A very precise measurement has small random errors, but just because a measurement is precise doesn't mean that it's accurate The accuracy of a measurement is a way of talking about the total error in our final result. An accurate measurement is very close to the true value. meaze@hep.knu.ac.kr 18th May 2004

The word "precision" is related to the random error distribution associated with a particular experiment or even with a particular type of experiment. Precision – how big the scatter of repeated measurements about the mean value – depends on the magnitude of random errors only The word "accuracy" is related to the existence of systematic errors—differences between laboratories, for instance. Accuracy – how close the value of a measurement is believed to be to the true value . For example, one could perform very precise but inaccurate timing with a high-quality pendulum clock that had the pendulum set at not quite the right length. meaze@hep.knu.ac.kr 18th May 2004

SYSTEMATIC ERROR Systematic errors – deviations between the mean of a large number of measured values and the true value – due to limitations of the measurement equipment or improper calibration – sometimes called “bias” – always present in measurements – must be estimated by the person making the measurements – can not be analyzed using statistics meaze@hep.knu.ac.kr 18th May 2004

Systematic error is an error inherent in the experimental set up which causes the results to be skewed in the same direction every time, i.e., always too large or always too small i.e., Systematic errors are those errors in a measurement that are regular and consistent in the sense that the measurements are consistently too large or too small, and usually by about the same amount for each measurement. One example of systematic error would be trying to measure the fall time of a ping pong ball to determine the acceleration due to gravity.  Air resistance would systematically reduce the measured acceleration, producing a systematic error.  meaze@hep.knu.ac.kr 18th May 2004

Systematic errors are usually much more difficult to detect and measure. We are often unaware of their existence. If one is measuring something whose value is already thought to be known, and consistently obtains a value that is significantly different, one is certainly led to suspect the presence of one or more systematic errors. However, if one is measuring something that has never been measured before, any systematic error present will not be obvious, and great care must be taken to investigate and eliminate every source of systematic error that one can think of. Experience and ingenuity are the valuable assets in such an investigation. Obviously sometimes things are overlooked and values published that later turn out to be wrong. meaze@hep.knu.ac.kr 18th May 2004

RANDOM ERROR Random errors – deviations between measured values and the mean value of a large number of repeated measurements – due to limitations of the measurement equipment or operator technique – can be accounted for by statistics through repetitions of the measurement meaze@hep.knu.ac.kr 18th May 2004

Random errors are errors which fluctuate from one measurement to the next. They yield results distributed about some mean value. They can occur for a variety of reasons. They may occur due to lack of sensitivity. For a sufficiently a small change an instrument may not be able to respond to it or to indicate it or the observer may not be able to discern it. They may occur due to noise. There may be extraneous disturbances which cannot be taken into account. They may be due to imprecise definition. They may also occur due to statistical processes. meaze@hep.knu.ac.kr 18th May 2004

MATHEMATICAL EXPRESSIONS In an experiment the quantity X    is measured, say N times. Results of the individual measurements are represented by a set of N numerical values Xi, i = 1, 2, ..., N. The true numerical value Xt of the measured quantity is approximated by the arithmetic mean of the measurements, if the errors are distributed according to the normal distribution. The average is given by The standard error of data in the sample is determined by the standard deviation, Both, the average and the standard deviation are quantities that have errors. The error of the average is We will report the result of the measurement in the form

If u represents the uncertainty in a measurement, which is the estimated error in the measurement, Random errors displace measurements in an arbitrary direction whereas systematic errors displace measurements in a single direction. Random errors are unavoidable and must be lived with. meaze@hep.knu.ac.kr 18th May 2004

EXAMPLE meaze@hep.knu.ac.kr 18th May 2004

Be careful! Wrong doesn't mean bad! CONCLUSIONS Every experimental result is subject to error. One can attempt to minimize errors but cannot eliminate them completely. Be careful!  Wrong doesn't mean bad! meaze@hep.knu.ac.kr 18th May 2004