Ej530a Error Curve
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Overview Keeping a lab notebook Writing research papers Dimensions & units Using figures (graphs) Examples of graphs Experimental error Representing error Applying statistics Overview Principles of microscopy Solutions & dilutions Protein assays Spectrophotometry Fractionation & centrifugation Radioisotopes which of these data points is the most reliable? and detection Error Representation and Curvefitting As far as the laws of mathematics normal error curve in analytical chemistry refer to reality, they are not certain; and as far as they are certain, they do not refer to according to adherents of ret the long-run result of this increase in government spending will be reality --- Albert Einstein (1879 - 1955) This article is a follow-up to the article titled "Error analysis and significant figures," which introduces important terms and concepts. The present article covers the which quantity is used to generate the error bars on a graph? rationale behind the reporting of random (experimental) error, how to represent random error in text, tables, and in figures, and considerations for fitting curves to experimental data. You might also be interested in our tutorial on using figures (Graphs). When to report random error Random error, known also as experimental error, contributes uncertainty to any experiment or observation that involves measurements. One must take
Error Bars On Four Different Data Points Are Shown Below.
such error into account when making critical decisions. When you present data that are based on uncertain quantities, people who see your results should have the opportunity to take random error into account when deciding whether or not to agree with your conclusions. Without an estimate of error, the implication is that the data are perfect. Random error plays such an important role in decision making, it is necessary to represent such error appropriately in text, tables, and in figures. When we study well defined relationships such as those of Newtonian mechanics, we may not require replicate sampling. We simply select enough intervals at which to collect data so that we are confident in the relationship. Connecting the data points is then sufficient, although it may be desirable to use error bars to represent the accuracy of the measurements. When random error is unpredictable enough and/or large enough in magnitude to obscure the relationship, then it may be appropriate to carry out replicate sampling and represent error in the figure. Representing experimental error The definitions of mean, standard deviation, and standard deviation of the mean were made in the previous article.
is called a "normal http://www.ruf.rice.edu/~bioslabs/tools/data_analysis/errors_curvefits.html error" or Gaussian curve. It is also sometimes called a "bell-shaped curve". The mathematical expression for this random distribution is: Where http://www.tissuegroup.chem.vt.edu/chem-ed/data/gaussian.html is the population standard deviation: and with µ being the population mean. This equation is plotted below. The dark vertical lines indicate the area under the curve from -1 to +1, and the lighter lines indicate the area under the curve for ±2. Related topics: data handling Top of Page Copyright © 2000 by Brian M. Tissue, all rights reserved.
the English-French mathematician de Moivre (in 1733), and later Laplace and Gauss. For the distribution corresponding to the curve we find names as: de Moivre distribution Gaussian distribution Gauss http://www.2dcurves.com/exponential/exponentialg.html Laplace distribution The Gaussian distribution can be proved (by the so-called Central Limit Theorem) http://ej530a.error.curve.repaircomputerhelp.org/ in the situation that each measurement is the result of a large amount of small, independent error sources. These errors have to be of the same magnitude, and as often positive as negative. When measuring a physical variable one tries to eliminate systematic errors, so that only accidental errors have to be taken error bars into account. In that case the measured values will spread around the average value, as a Gauss curve. It can be proved that in the case when the average value of a measured value is the 'best value', a Gaussian distribution holds. The 'best value' is here defined as that value, for which the chance on subsequent measurements is maximal 1). Because in general an estimation of errors error bars on is rather rough, the distribution to be used has not to define the error very precise. More important is that the distribution is easy to work with. And the Gaussian distribution has that quality in many situations. Some real life examples of the Gauss distribution: distribution of the length of persons (given the sex) distribution of the weight of machine packed washing powder distribution of the diameter of machine made axes Taking the definition of the standard deviation 2) it can be seen that σ is the standard deviation in the Gauss distribution of the form: The points of inflection are situated at x = ± σ. For this distribution about two of the three measurements has a distance less than σ from the maximum value. And about one of the twenty measurements has a distance of more than 2σ. Another interesting quality of the Gauss curve is that it is the only function which remains unchanged for a Fourier transform. Because its form the curve is also called the bell curve. The curve can be signed to the flipped Gauss curve. notes
1) Squires 1972 p. 40. 2) The standard deviation of a distribution function f(x) is defined as: