A textbook on the method of least squares8th ed., rev . he probable error r, consider the case oftwo sets of observations made with different degrees of accu-racy. Let the measure of precision of the first be /ilt and ofthe second /z2; then, from equation (2), the probability of errorsin the first set will be represented by a curve whose equation is y = *e-h?*z, and for the second set by a curve y = —be—ti**, in which dx is the constant difference between two consecutiveerrors. Now, suppose that the second set is twice as preciseas the first, so that h^ = h, and /z2 = 2k; then
A textbook on the method of least squares8th ed., rev . he probable error r, consider the case oftwo sets of observations made with different degrees of accu-racy. Let the measure of precision of the first be /ilt and ofthe second /z2; then, from equation (2), the probability of errorsin the first set will be represented by a curve whose equation is y = *e-h?*z, and for the second set by a curve y = —be—ti**, in which dx is the constant difference between two consecutiveerrors. Now, suppose that the second set is twice as preciseas the first, so that h^ = h, and /z2 = 2k; then the equations will be y = ahc-h*x* and y = 2ahe-*h*x*, in which a represents the constant ir~^dx. The curves corre- 68 THE PRECISION OF OBSERVATIONS. IV. sponding to these equations are given in Fig. 6; XB^A^^Xbeing the one for the set of observations whose measure ofprecision is hl or h, and XB2A2B2X the one for the set whosemeasure of precision is h2, or 2/1. These curves show at aglance the relative probabilities of corresponding errors in the. two sets : thus the probability of the error o is twice as muchin the second as in the first set; the probability of the errorOPj is nearly the same in each; while the probability of anerror twice as large as OP, is much smaller in the second thanin the first set. Now, if the lines PtBlt P2B2 be drawn so thatthe areas PlBIA1BIPI and P2B2A2B2P2 are respectively one-half of the total areas of their corresponding curves, the lineOPt will be the probable error of an observation in the first set,and 0P2 the probable error of one in the second set. Repre-senting these by the letters r, and r2, there must be in eachcase the constant relation h,r, = , h2r2 = ; § 63. THE PROBABLE ERROR. 69 and, since k2 is twice hx, it follows that r2 must be one-halfof r,. The probable error, then, serves to compare the precision ofobservations equally as well as measures of precision. Thesmaller the probable error, the more precise are the
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