Abstract
The relationship between Root Mean Square (RMS) error and probable error was investigated under the condition of normal distribution with independent x, y and z error components. In the ideal case of no bias and equal standard deviations (in the case of two or three dimensions), the ratios of 90% probable error to RMS error are 1.645 (one dimensional normal distribution), 2.144 (two dimension) and 2.333 (three dimension) . Similarly the ratios for 50% probable error are 0.674 (one dimension), 1.177 (two dimension) and 1.500 (three dimension) . The bias within one standard deviation causes the variation of the ratio of 90% probable error to RMS error by the value of 0.0295 (2% of the above ratio) in one dimensional case, 0.0433 (2%) in two dimensional case, and 0.1771 (8%) in three dimensional case. Similarly, the different values of standard deviation within 2 times each other causes the variation of the ratios of 90% probable error to RMS error by the value within 0.3104 (14% of the above ratio) in two dimensional case and 0.2404 (10%) in three dimensional case.
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