Title 107 Theory of Errors.pdf ✅

MSTC 107: Theory of Errors 1. Theory of Probability The perfect physical quantity measurement is measured using a most probable value and a tolerance value. In this discussion, the errors are based on 50% probability. 1.1. Most Probable Value The most probable value of a measurement, especially when subjected to multiple trials, is MPV = x = ∑ x n 1.2. Probable Errors A probable error in a measurement is developed from confidence intervals based on standard errors and standard deviations. • Residual is the difference between the actual measurement and the most probable value. v = x − x • Variance – The sample variance is used since multiple measurement trials are considered sample measurements. s 2 = ∑ v 2 n − 1 • Standard Deviation – It is the square root of the variance, i.e., s or s = √ ∑ v 2 n − 1 • Standard Error – A statistical tool measures how much a sample mean differs from a population mean. SE = s √n • Probable Error of any Measurement PEs = ±0. 6745s • Probable Error of the Mean PEm = ±0. 6745(SE)