1.
Discuss the difference between an error and a residual.
Definition of error is the difference between the observation and its true value.
𝜀𝜀 = 𝑙𝑙(𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜) − 𝐿𝐿(𝑜𝑜𝑜𝑜𝑡𝑡𝑜𝑜 𝑜𝑜𝑜𝑜𝑙𝑙𝑡𝑡𝑜𝑜)
whereas a residual, which is defined is the difference between the most probable value
and the observation.
̅
𝑜𝑜𝑖𝑖 = 𝑙𝑙 (𝑚𝑚𝑜𝑜𝑜𝑜𝑜𝑜
𝑝𝑝𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑙𝑙𝑜𝑜 𝑜𝑜𝑜𝑜𝑙𝑙𝑡𝑡𝑜𝑜) − 𝑙𝑙𝑖𝑖 (𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜)
2.
Distinguish between cumulative and accidental errors.
Cumulative errors are systematic errors which accumulate. It result from factors that
comprise the “measuring system” and include the environment, instrument, and
observer. Conditions producing systematic errors conform to physical laws that can be
modeled mathematically. Thus, if the conditions are known to exist and can be observed,
a correction can be computed and applied to observed values.
On the other hand, accidental errors are random errors caused by factors beyond the
control of the observer. They are those that remain in measured values after mistakes
and systematic errors have been eliminated. There is no absolute way to compute or
eliminate them. Such errors obey the laws of probability.
3.
A distance AB is observed repeatedly using the same equipment and
procedures, and the results, in meters, are listed in Problems 3.6. Calculate (a)
the line’s most probable length, (b) the standard deviation and (c) the
standard deviation of the mean for each set of results.
65.401, 65.400, 65.402, 65.396, 65.406, 65.401, 65.396, 65.401, 65.405, and 65.404
3.1 𝑚𝑚𝑜𝑜𝑜𝑜𝑜𝑜 𝑋𝑋�
=
∑ 𝑋𝑋𝑖𝑖
𝑁𝑁
65.401 + 65.400 + 65.402 + 65.396 + 65.406 +
�
= 65.401 + 65.396 + 65.401 + 65.405 + 65.404
10
�
= 65.401(𝑚𝑚)
∑(𝑋𝑋𝑖𝑖 −𝑋𝑋�)2
3.2 ±𝜎𝜎 = ±�
𝑁𝑁−1
�
(65.401 − 65.401)2 +
⃓
⃓
⃓
(65.396 − 65.401)2 +
⃓
⃓
(65.396 − 65.401)2 +
⃓
⃓
⃓
= ±⃓
⎷
(65.400 − 65.401)2 +
(65.406 − 65.401)2 +
(65.401 − 65.401)2 +
(65.404 − 65.401)2
(10 − 1)
1
(65.402 − 65.401)2 +
(65.401 − 65.401)2 +
(65.405 − 65.401)2 +
= ±0.003(𝑚𝑚)
3.3 ±𝜎𝜎𝑋𝑋� = ±
4.
𝜎𝜎
√𝑁𝑁
=±
0.003
√10
= ±0.001(𝑚𝑚)
Line AD is observed in three sections, AB, BC, and CD, with lengths and
standard deviations as listed below. What is the total length AD and its
standard deviation?
4.1. 𝑨𝑨𝑨𝑨 = 𝟐𝟐𝟐𝟐𝟐𝟐. 𝟓𝟓𝟓𝟓 ± 𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎; 𝑨𝑨𝑩𝑩 = 𝟐𝟐𝟎𝟎𝟔𝟔. 𝟗𝟗𝟗𝟗 ± 𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎;
𝑩𝑩𝑪𝑪 = 𝟒𝟒𝟐𝟐𝟐𝟐. 𝟔𝟔𝟓𝟓 ± 𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎
����
���� + ����
𝐴𝐴𝐴𝐴 = ����
𝐴𝐴𝐴𝐴 + 𝐴𝐴𝐵𝐵
𝐵𝐵𝐴𝐴 = 236.57 + 608.99 + 426.87 = 1272.43(𝑓𝑓𝑜𝑜)
2
2
2
±𝜎𝜎𝐴𝐴𝐴𝐴 = ±�𝜎𝜎𝐴𝐴𝐴𝐴
+ 𝜎𝜎𝐴𝐴𝐵𝐵
+ 𝜎𝜎𝐵𝐵𝐴𝐴
= �(0.01)2 + (0.01)2 + (0.01)2 = ±0.02(𝑓𝑓𝑜𝑜)
���� = 1272.43 ± 0.02(𝑓𝑓𝑜𝑜)
𝐴𝐴𝐴𝐴
4.2. 𝑨𝑨𝑨𝑨 = 𝟐𝟐𝟔𝟔𝟔𝟔. 𝟗𝟗𝟔𝟔𝟎𝟎 ± 𝟎𝟎. 𝟎𝟎𝟎𝟎𝟐𝟐𝟎𝟎; 𝑨𝑨𝑩𝑩 = 𝟎𝟎𝟐𝟐𝟓𝟓𝟒𝟒. 𝟔𝟔𝟐𝟐𝟓𝟓𝟎𝟎 ± 𝟎𝟎. 𝟎𝟎𝟎𝟎𝟐𝟐𝟎𝟎;
𝑩𝑩𝑪𝑪 = 𝟐𝟐𝟓𝟓𝟒𝟒𝟐𝟐. 𝟐𝟐𝟓𝟓𝟐𝟐𝟎𝟎 ± 𝟎𝟎. 𝟎𝟎𝟎𝟎𝟓𝟓𝟎𝟎
����
���� + ����
𝐴𝐴𝐴𝐴 = ����
𝐴𝐴𝐴𝐴 + 𝐴𝐴𝐵𝐵
𝐵𝐵𝐴𝐴 = 688.980 + 1274.865 + 2542.373 = 4506.218(𝑚𝑚)
2
2
2
±𝜎𝜎𝐴𝐴𝐴𝐴 = ±�𝜎𝜎𝐴𝐴𝐴𝐴
+ 𝜎𝜎𝐴𝐴𝐵𝐵
+ 𝜎𝜎𝐵𝐵𝐴𝐴
= �(0.003)2 + (0.003)2 + (0.005)2 = ±0.006(𝑚𝑚)
���� = 4506.218 ± 0.006(𝑚𝑚)
𝐴𝐴𝐴𝐴
5.
Specifications for observing angles of an n-sided polygon limit the total
angular misclosure to E. How accurately must each angle be observed for the
following values of n and E?
𝑜𝑜𝑡𝑡𝑚𝑚 𝑜𝑜𝑓𝑓 𝑜𝑜ℎ𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑎𝑎𝑙𝑙𝑜𝑜 = 𝐴𝐴𝑜𝑜𝑎𝑎𝑙𝑙𝑜𝑜1 + 𝐴𝐴𝑜𝑜𝑎𝑎𝑙𝑙𝑜𝑜2 + ⋯ 𝐴𝐴𝑜𝑜𝑎𝑎𝑙𝑙𝑜𝑜𝑛𝑛
±𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 = ±�𝐸𝐸 2 + 𝐸𝐸 2 + 𝐸𝐸 2 + ⋯ = ±�𝑜𝑜𝐸𝐸 2
𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 : 𝑜𝑜ℎ𝑜𝑜 𝑝𝑝𝑜𝑜𝑜𝑜𝑝𝑝𝑜𝑜𝑎𝑎𝑜𝑜𝑜𝑜𝑜𝑜𝑝𝑝 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑡𝑡𝑚𝑚 𝑜𝑜𝑓𝑓 𝑜𝑜ℎ𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑎𝑎𝑙𝑙𝑜𝑜
𝑜𝑜: 𝑜𝑜ℎ𝑜𝑜 𝑜𝑜𝑡𝑡𝑚𝑚𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑓𝑓 𝑚𝑚𝑜𝑜𝑜𝑜𝑜𝑜𝑡𝑡𝑜𝑜𝑜𝑜𝑚𝑚𝑜𝑜𝑜𝑜𝑜𝑜s
𝐸𝐸: 𝑜𝑜ℎ𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜𝑝𝑝𝑜𝑜𝑚𝑚 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑓𝑓 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑡𝑡𝑚𝑚𝑜𝑜𝑜𝑜𝑜𝑜
5.1 ±
𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠
5.2 ±
𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠
√𝑛𝑛
√𝑛𝑛
= ±𝐸𝐸 = ±
= ±𝐸𝐸 = ±
8′′
√8
= ±2.8′′
12′′
√16
= ±3.0′′
2