Random Error Calculation Physics
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in measuring the time required for a weight to fall to the floor, a random error will occur when an experimenter attempts to push a button that starts a timer simultaneously with the release of the weight. errors in measurement physics class 11 If this random error dominates the fall time measurement, then if we repeat the
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measurement many times (N times) and plot equal intervals (bins) of the fall time ti on the horizontal axis against the error in physics definition number of times a given fall time ti occurs on the vertical axis, our results (see histogram below) should approach an ideal bell-shaped curve (called a Gaussian distribution) as the number of measurements N becomes
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very large. The best estimate of the true fall time t is the mean value (or average value) of the distribution: átñ = (SNi=1 ti)/N . If the experimenter squares each deviation from the mean, averages the squares, and takes the square root of that average, the result is a quantity called the "root-mean-square" or the "standard deviation" s of the distribution. It measures the random error or the error calculation formula statistical uncertainty of the individual measurement ti: s = Ö[SNi=1(ti - átñ)2 / (N-1) ].
About two-thirds of all the measurements have a deviation less than one s from the mean and 95% of all measurements are within two s of the mean. In accord with our intuition that the uncertainty of the mean should be smaller than the uncertainty of any single measurement, measurement theory shows that in the case of random errors the standard deviation of the mean smean is given by: sm = s / ÖN , where N again is the number of measurements used to determine the mean. Then the result of the N measurements of the fall time would be quoted as t = átñ ± sm. Whenever you make a measurement that is repeated N times, you are supposed to calculate the mean value and its standard deviation as just described. For a large number of measurements this procedure is somewhat tedious. If you have a calculator with statistical functions it may do the job for you. There is also a simplified prescription for estimating the random error which you can use. Assume you have measured the fall time about ten times. In this case it is reasonable toof causes of random errors are: electronic noise in the circuit of an electrical instrument, irregular changes in the heat loss rate from a solar collector due to changes in the wind. Random errors often have a Gaussian normal distribution (see Fig. 2). In such cases
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statistical methods may be used to analyze the data. The mean m of a number
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of measurements of the same quantity is the best estimate of that quantity, and the standard deviation s of the measurements shows the accuracy systematic error calculation of the estimate. The standard error of the estimate m is s/sqrt(n), where n is the number of measurements. Fig. 2. The Gaussian normal distribution. m = mean of measurements. s = standard deviation of measurements. 68% of the http://felix.physics.sunysb.edu/~allen/252/PHY_error_analysis.html measurements lie in the interval m - s < x < m + s; 95% lie within m - 2s < x < m + 2s; and 99.7% lie within m - 3s < x < m + 3s. The precision of a measurement is how close a number of measurements of the same quantity agree with each other. The precision is limited by the random errors. It may usually be determined by repeating the measurements. Systematic Errors Systematic http://www.physics.umd.edu/courses/Phys276/Hill/Information/Notes/ErrorAnalysis.html errors in experimental observations usually come from the measuring instruments. They may occur because: there is something wrong with the instrument or its data handling system, or because the instrument is wrongly used by the experimenter. Two types of systematic error can occur with instruments having a linear response: Offset or zero setting error in which the instrument does not read zero when the quantity to be measured is zero. Multiplier or scale factor error in which the instrument consistently reads changes in the quantity to be measured greater or less than the actual changes. These errors are shown in Fig. 1. Systematic errors also occur with non-linear instruments when the calibration of the instrument is not known correctly. Fig. 1. Systematic errors in a linear instrument (full line). Broken line shows response of an ideal instrument without error. Examples of systematic errors caused by the wrong use of instruments are: errors in measurements of temperature due to poor thermal contact between the thermometer and the substance whose temperature is to be found, errors in measurements of solar radiation because trees or buildings shade the radiometer. The accuracy of a measurement is how close the measurement is to the true value of the quantity being measured. The accuracy of measurements is often reduced by systematic errors, which are difficult to detect even for experienced research workers.
Taken from R. H. B. Exell, www.jgsee.kmutt.Use of Errors Determination of Errors Experimental Errors Random Errors Distribution Curves Standard Deviation Systematic Errors Errors in Calculated Quantities Rejection of Readings MEASUREMENT All science is concerned with measurement. This fact requires that we have standards of measurement. Standards In http://webs.mn.catholic.edu.au/physics/emery/measurement.htm order to make meaningful measurements in science we need standards of commonly measured quantities, such as those of mass, length and time. These standards are as follows: 1. The kilogram is the mass of a cylinder of platinum-iridium alloy kept at the International Bureau of Weights and Measures in Paris. By 2018, however, this standard may be defined in terms of fundamental constants. For further information read: http://www.nature.com/news/kilogram-conflict-resolved-at-last-1.18550 . 2.The metre is defined as the length of the path travelled error calculation by light in a vacuum during a time interval of 1/299 792 458 of a second. (Note that the effect of this definition is to fix the speed of light in a vacuum at exactly 299 792 458 m·s-1). 3.The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. It is necessary for all such standards to be error analysis physics constant, accessible and easily reproducible. Top SI Units Scientists all over the world use the same system of units to measure physical quantities. This system is the International System of Units, universally abbreviated SI (from the French Le Système International d'Unités). This is the modern metric system of measurement. The SI was established in 1960 by the 11th General Conference on Weights and Measures (CGPM, Conférence Générale des Poids et Mesures). The CGPM is the international authority that ensures wide dissemination of the SI and modifies the SI as necessary to reflect the latest advances in science and technology. Thus, the kilogram, metre and second are the SI units of mass, length and time respectively. They are abbreviated as kg, m and s. Various prefixes are used to help express the size of quantities – eg a nanometre = 10-9 of a metre; a gigametre = 109 metres. See the table of prefixes below. Table 1. SI prefixes Factor Name Symbol 1024 yotta Y 1021 zetta Z 1018 exa E 1015 peta P 1012 tera T 109 giga G 106 mega M 103 kilo k 102 hecto h 101 deka da Factor Name Symbol 10-1 deci d 10-2 centi c 10-3 milli m 10-6 micro µ 10-9 nano n 10-12 pico p 10-15 femto f 10-18 atto a 10-21 zepto z 10-24 yocto y Top Fundamental and Derived Quantities Physical quantities are not generally independ
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