# Principalformulas

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Principal Formulas in Part | Notation (Chapter 2) (Measured value of x) = x,., = Ox, (p- 13) where Xpesy = Dest estimate for x, ox = uncertainty or error in the measurement. : : Ox Fractional uncertainty = : (p. 28) |xbest| Propagation of Uncertainties (Chapter 3) If various quantities x, ..., w are measured with small uncertainties ox, ..., ow, and the measured values are used to calculate some quantity g, then the uncertainties in x, ..., w cause an uncertainty in g as follows: If g is the sum and difference, g = x + -+ + z (u + --- + w), then = B2+ -+ (82)% + (Bu)® + - - + (8w)? 5q for independent random errors; <= &b +---+86+06u+- -+ ow always. (p- 60) If g is th duct and tient, g = , th q is the product and quotient, g = ~—————"—, then ( Ox\2 0z\2 ou\2 ow\2 B (_) + o + (—) + (———) + o + (——) X Z U w oq J for independent random errors; "1' Ox 0z Oou ow < 4 o4 = = 44+ |x] o] |y W] ) always. (p. 61) If g = Bx, where B i1s known exactly, then = |B|éx. (p- 54) If g is a function of one variable, g(x), then d 5g = -fi 5x. (p. 65) If g is a power, g = X", then o o 1 = | =, (p. 66)
If g is any function of several variables x, . . ., z, then 09 . \? 9 o\ Sa = 5 ...+ 28 . 1 \/(ax x) (az Z) (p- 75) (for independent random errors). Statistical Definitions (Chapter 4) If x,,..., xy denote N separate measurements of one quantity x, then we define: 1 N X = NZ X; = mean, (p. 98) i=1 1 o, = \/ N1 Z(x,- X)* = standard deviation, or SD (p. 100) oz = "x = standard deviation of mean, or SDOM. (p. 102) N The Normal Distribution (Chapter 5) For any limiting distribution f(x) for measurement of a continuous variable x: f(x) dx = probability that any one measurement will give an answer between x and x + dx; (p- 128) b f f(x) dx = probability that any one measurement will ¢ give an answer between x = a and x = b; (p. 128) f f(x)dx = 1 is the normalization condition. (p- 128) The Gauss or normal distribution is Gy ,(X) = g~ X207 (p. 133) oN2T where X = center of distribution = true value of x = mean after many measurements, o = width of distribution = standard deviation after many measurements. The probability of a measurement within ¢ standard deviations of X is Prob(within to) = —2%2 4- = normal error integral; (p. 136) 1 t = . \2m J - in particular Prob(within 10) = 68%.
Principal Formulas in Part Il Weighted Averages (Chapter 7) If x;, ..., xy are measurements of the same quantity x, with known uncertainties oy, - .., Oy, then the best estimate for x is the weighted average DWW x; Xwav ZVLI (p 175) where the weight w, = 1/0°. Least-Squares Fit to a Straight Line (Chapter 8) If (x;,y1), ..., (xpn, yy) are measured pairs of data, then the best straight line y = A + Bx to fit these N points has A = [(zxiz)(zyi) (in)(zxiyi)]/A, B [N(zxiyi) (zxi) (zyz')]/A» where A = N(le-z) - (zxi)z- (p. 184) Covariance and Correlation (Chapter 9) The covariance o,, of N pairs (x;, y;), ..., (X, Yy) 18 1 _ Ty = X,Z(xi )0 ). (p. 212) The coefficient of linear correlation is 1 = X)W (Vv. V Xy z(xz x)(yl y) (p 217) 7.0y Xy D22y Y Values of r near 1 or 1 indicate strong linear correlation; values near O indicate little or no correlation. (For a table of probabilities for 7 see Appendix C.) r: Binomial Distribution (Chapter 10) If the probability of "success" in one trial 1s p, then the probability of v successes in n trials 1s given by the binomial distribution v Prob(v successes in n trials) = B, ,(v) = T Y p" (1 p)*~ % ' v)! vi(n (p- 230)
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