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The replacement of nīy (n-1) us called Bessel's correction. See the books by Topping, Parratt, Beers, Barford, and Pugh-Winslow. Mathematical statistics texts may be consulted for an explanation of equation 5.5. The distinctionīetween the two is mainly important for small samples. 5.3 and 5.6 become more nearly identical as n gets large. When this factor is applied to the root mean square deviation, the result is simply to We must always be content with a finite sample, but we would like to use it toĮstimate the dispersion of the parent distribution. In statistical theory one speaks of the parentĭistribution, an infinite set of measurements of which our finite sample is but a subset. Ideally we want huge samples, for the larger the sample, the more nearly the Had we taken more data, we would expect slightly different answers both the meanĪnd the dispersion depends on the size of the sample. The dispersion measures listed in the last section described the dispersion of the data This is 1.6949σ.ĥ.5 ESTIMATES OF DISPERSION OF THE "PARENT" DISTRIBUTION Of the mean will include 90% of the data values. RELIABLE ERROR (Def.) A range within one reliable error on either side On either side of the mean will include 50% of the data values. PROBABLE ERROR (P.E.) (Definition) A range within one probable error (This is not a definition.)Ī range within two standard deviations will include 95% of the data values. Mean will include approximately 68% of the data values. STANDARD DEVIATION (or STANDARD ERROR, σ): A range within one standard deviation on either side of the Statisticians have devised better measures of "width" of Gaussian curves by specifyingĪ range of values of x which include a specified fraction of the measurements. This is not very useful in statistical studies. Then the "width at half height" is (x 2 - x 1). Measured by finding two points x 1 and x 2 such that f(x 1) = f(x 2) = f()/2. Sometimes define the "width" of such peaked curves by the "width at half height." This is Gaussians do differ in how much f(x) decreases for a given value of (x - ). The total width, or spread, of the Gaussian curve is infinite, as the equation shows. Furthermore, when one must deal with an unknown distri-īution, it is usually assumed to be Gaussian until contrary evidence is found. The Gaussian distribution is so common that much of the terminology of statistics andĮrror analysis has been built upon it. The shape of the Gaussian shows it to be symmetric about its highest value, and this 5.4 has been accurately drawn to illustrate this curve. σ is the standard deviation.ĭistributions which conform to this equation are called Gaussian, or Where x is a measurement, is the mean, and f(x) is the ordinate of the distributionĬurve for that value of x. The distributions encountered in physics often have a mathematical shape given Or, simply the square root of the mean squareĥ.4 DISPERSION MEASURES APPROPRIATE TO GAUSSIAN DISTRIBUTIONS
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ROOT MEAN SQUARE DEVIATION The square root of the average of the The average of the sum of the squares of the (usually justĪVERAGE DEVIATION, abbreviated lower case, a. Some commonly used measures of dispersion are listed forĪVERAGE DEVIATION FROM THE MEAN. The difference between a measurement and the mean of its distribution is called theĭEVIATION (or VARIATION) of that measurement. The value at which the peak of the distribution curve occurs.) MODE The most frequent value in a set of measurements. The middle value of a set of measurements ranked in numerical The reciprocal of the average of the reciprocals of The nth root of the product of n positive Of the measurements divided by the number of measurements. (or simply the MEAN, or the AVERAGE): The sum Some of the "measures of central tendency" commonly used are listed here forĪRITHMETIC MEAN. The mathematicalĭiscipline of statistics has developed systematic ways to do this. We canĭescribe the measurement and its uncertainty by just a few numbers. The distribution curves by measures of dispersion (spread), skewness, etc. One value (some kind of average), so also we can represent the shape of One can often guess the shape of the curve, even with a finite set of values, especially Such aĬurve is called an error distribution curve. The tops of the bars are connected with a smooth curve. Number of values is very large, and a bar graph (Fig. As always, one proceeds on the basis of reasonableĬonsider a large number of repeated measured values of a physical quantity. In practice, one must deal with a finite set of values, so the nature of their distribution Some of the methods for accurately describing the nature of measurement distributions. Intuitive way, without inquiring into the nature of the scatter. Up to this point, the discussion has treated the "scatter" of measurements in an