The Theory of Errors and Method of Least Squares
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1892 edition. Excerpt: ...and r, it is necessary to compare the difference d with the relative probable error tj (r + rj), Art. 96. If d is small enough to be regarded as a relative accidental error, it is safe to make the assumption and combine the determinations in the manner mentioned above. As an example, let us suppose that a certain angle has been determined by a theodolite as 240 13' 36" 3."i, and that a second determination made with a surveyor's transit 240 13'24" I3."8. In this case r, = 3.1, r2 = 13.8 and d= 12. It is obvious that a relative accidental error as great as d may reasonably be expected. (In fact the relative probable error is 14.1; and, by Table II, the chance that the accidental error should be at least as great as 12 is about.57.) We may therefore assume tha'. there is no relative systematic error, and combine the determinations with weights having the inverse ratio of the squares of the probable errors. This ratio will be found, in the present c.xse, tc be about 20: 1, and the corresponding weighted mean found by adding.fa of the difference to the first value, is 24 13' 35."43. 100. It appears doubtful at first that the value given by the VII. CONCORDANT DETERMINATIONS. 81 theodolite can be improved by combining with it the value given by the inferior instrument. The propriety of the above process becomes more apparent, however, if we imagine the first determination to be the mean of twenty observations made with the theodolite; a single one of these observations will then have the same weight and the same probable error as the second determination. Now the discrepancy of this new determination from the mean is such as we may expect to find in a new single observation with the theodolite. We are...
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