Theory of Errors
1. A distance AB is observed repeatedly using the same equipment and procedures, and the result, in meters, are listed as follows: 125.474, 125.471, 125.478, 125.476, 125.472, 125.473, 125.480, 125.478, 125.478, and 125.474. Calculate a) the line's most probable length, b) the standard deviation, and c) the standard deviation of the mean.
2. An angle is observed repeatedly using the same equipment and procedures. Calculate a) the angle's most probable value, b) the standard deviation, and c) the standard deviation of the mean.
130°32'36'', 130°32'42'', 130°32'44'', and 130°32'38''.
3. A field party is capable of making taping observations with a standard deviation of ±0.015 ft per 100-ft tape length. What standard deviation would be expected in a distance of 300 ft taped by this party?
4. Determine the weighted mean for the following angles:
a) 168°43'26'', wt 3; 168°43'16'', wt 1; 168°43'22'', wt 4;
b) 36°58'32'' ± 4''; 36°58'28'' ± 3''; 36°58'42'' ± 6''; 36°58'30'' ± 2''
5. What is the area of a rectangle field and its estimated error for the following recorded values:
108.014 ± 0.003 m by 287.010 ± 0.005 m
6. A certain angle was measured ten times by observer A with the following results, all measurements being equally precise:
74°38'18", 20", 15", 21", 24", 16", 22", 17", 19", 13" (The degrees and minutes remained constant for each observation.)
The same angle was measured under the same conditions by observer B with the following results:
74°38'10", 21", 25", 08", 15", 20", 28", 11", 18", 24"
Determine the standard deviation for each observer and relative weightings.
7. Determine relative weights and perform a weighted adjustment (to the nearest second) for angles A, B, and C of plane triangle, given the following four observations for each angle:
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Angle A
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Angle B
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Angle C
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38°47'58''
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71°22'26''
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69°50'04''
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38°47'44''
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71°22'22''
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69°50'16''
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38°48'12''
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71°22'12''
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69°50'30''
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38°48'02''
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71°22'12''
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69°50'10''
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