Theory of Errors in Observations

1. Theory of Errors in Observations Chapter 3 (continued)

2. 50, 90 and 95 Percent Errors The 50% error or probable error establishes the limits within an observation has the same chance of falling within the limits or outside of them. The 90 and 95% errors are used to specify precisions required for surveying projects.

3. Error of a Sum • Independently observed observations • Measurements made using different equipment, under different environmental conditions, etc. Where E represents any specified percentage error And a, b and c represent separate, independent observations

4. Example: • A line is observed in three sections with the lengths 1086.23 ± 0.05 ft, 569.08 ± 0.03 ft and 863.19 ± 0.04 ft. Compute the total length and standard deviation for the three sections. • Solution: Probable length = 2,518.50 ± 0.07 ft

5. Error of a Series • Similar Quantities (Like Measurements) • Measurements taken by the same equipment and under the same environmental conditions Where E represents the error in each individual observation and n is the number of observations

6. Example: • A field party is capable of making taping observations with a standard deviation of ± 0.015 ft per 100-ft tape length. What total standard deviation would be expected in a distance of 500 ft taped by this party? n = number of tape applications = 500’/100’ = 5 Probable length = 500.00 ± 0.03 ft

7. Error in a Product Ea and Eb are the respective errors in the sides A and B. A -Ea B +Ea - Eb +Eb

8. Example: • A rectangular plot of land is surveyed and the following measurements are recorded: 2245.68 ± 0.12 ft by 664.21 ± 0.06 ft. What is the area of the plot in acres and its expected error in square feet? Area = (2245.68)(664.21) = 1,491,603.11 sf Area in acres = 1,491,603.11 ÷ 43,560 = 34.24 acres

9. Error of the Mean E is the specified percentage error of a single observation and n is the number of observations This equation shows that the error of the mean varies inversely as the square root of the number of repetitions. In order to double the accuracy of a set of measurements you must take four times as many observations.

10. Weights of Observations Precise observations should be weighted more heavily than less precise observations. MW is the weighted mean, W is the Weight assigned to each measurement and M is the value of each measurement. An equation can be deduced from this proportionality which computes the relative weight of measurements based on their precision.

11. Weights of Observations (con.) These equations can be used to determine the relative weights of measurements based on their standard errors.