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# Accuracy: Random and Systematic Errors

Accuracy of Numerical Measurements and Analyses: Random and Systematic Errors

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### Accuracy: Random and Systematic Errors

1. 1. Accuracy of Numerical Results Random and Systematic Errors October 2010
2. 2. <ul><li>Introduction </li></ul><ul><li>Random Errors </li></ul><ul><li>Systematic Errors </li></ul><ul><li>Propagation of Errors </li></ul>Accuracy of Numerical Results
3. 3. Introduction <ul><li>Numerical results or analyses are an integral aspect of most research papers. </li></ul><ul><li>To ensure accuracy , and therefore reliability and integrity of the results: </li></ul><ul><li>Take into account experimental conditions or the method of data collection </li></ul><ul><li>Determine sources of error (random or systematic) </li></ul><ul><li>Errors are inherent to all data, and are distinct from mistakes or blunders !! </li></ul><ul><li>Examine and include contribution to errors from all sources </li></ul><ul><li>Assign appropriate errors to measured or derived quantities </li></ul>
4. 4. <ul><li>Introduction </li></ul><ul><li>Random Errors </li></ul><ul><li>Systematic Errors </li></ul><ul><li>Propagation of Errors </li></ul>Accuracy of Numerical Results
5. 5. Random Errors - I <ul><li>Simultaneous measurements of the same quantity: values distributed accordingly to the probability of occurrence (probability distribution) </li></ul><ul><li>Sources of random errors </li></ul><ul><ul><li>Statistical effects </li></ul></ul><ul><ul><li>Limitations in the measurement process </li></ul></ul><ul><li>Common probability distributions </li></ul><ul><li>Different forms of binomial distribution (low probability of occurrence/success) </li></ul><ul><ul><li>Small number of measurements or sample size ( Poisson ) </li></ul></ul><ul><ul><li>Large number of measurements or sample size ( Gaussian ) </li></ul></ul>
6. 6. Random Errors - II <ul><li>The distributions associated with the broad categories of random errors are: </li></ul><ul><li>Poisson distribution </li></ul><ul><ul><li>Small sample size </li></ul></ul><ul><ul><li>Discrete and asymmetric </li></ul></ul><ul><li>Gaussian/normal distribution </li></ul><ul><ul><li>Large sample size </li></ul></ul><ul><ul><li>Continuous and symmetric </li></ul></ul>-> Value Probability density -> Mean A typical Gaussian distribution indicating the probability of occurrence for a given value. μ indicates the mean value, while σ denotes the standard deviation. For an ideal Gaussian, 68.2% of the values would lie between 1 σ of the mean value , 95.4% between 2 σ , and 99.7% between 3 σ , as indicated in the above figure.
7. 7. <ul><li>Introduction </li></ul><ul><li>Random Errors </li></ul><ul><li>Systematic Errors </li></ul><ul><li>Propagation of Errors </li></ul>Accuracy of Numerical Results
8. 8. Systematic Errors - I <ul><li>Systematic errors tend to bias measured values in a pre-determined manner </li></ul><ul><li>Possible sources of systematic errors </li></ul><ul><ul><li>Imperfections in the measuring equipment </li></ul></ul><ul><ul><li>Incorrect use of apparatus or implementation of techniques </li></ul></ul><ul><li>Isolating/reducing systematic errors </li></ul><ul><ul><li>Examine and analyze measurement apparatus and techniques </li></ul></ul><ul><ul><li>Careful practices in the implementation of research </li></ul></ul>
9. 9. Systematic Errors - II <ul><li>Similar experiments performed at </li></ul><ul><li>different laboratories with varying </li></ul><ul><li>conditions and equipment have </li></ul><ul><li>unique associated systematic errors </li></ul><ul><li>If two different experiments report </li></ul><ul><li>results about the same quantity </li></ul><ul><li>which are variance with each </li></ul><ul><li>other, this may be due to incorrect </li></ul><ul><li>or incomplete treatment of </li></ul><ul><li>systematic errors in one or both. </li></ul>Systematic errors lead to a difference in the actual and measured values. In this simple example, the error can be characterized in terms of a difference in the offset (values at zero) and slope of the two lines.
10. 10. <ul><li>Introduction </li></ul><ul><li>Random Errors </li></ul><ul><li>Systematic Errors </li></ul><ul><li>Propagation of Errors </li></ul>Accuracy of Numerical Results
11. 11. Propagation of Errors <ul><li>Isolate and quantify all sources of random and systematic errors </li></ul><ul><li>Combine all sources of error: cumulative error on the measured quantity </li></ul><ul><li>Different procedure based on whether measured quantities are dependent or independent of each other </li></ul><ul><li>Reliability of quoted values depends on rigorous and thorough error analysis </li></ul><ul><li>Accurate values , with appropriate error bars , ensure sound results , which are respected by peers </li></ul>
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