3B. Error Theory - Part 2

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Apr 3, 2024

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Error Theory © R.L Wood and M.E. Mohammadi, 2021 Page 1 of 7 Error Theory – Part 2 Lesson Objectives: 1) Evaluate the measurement precision and calculate 50% , 90% , and 95% errors . 2) Calculate the propagation of the errors . 3) Describe the application of error theory in surveying and geomatics projects. Background Reading: 1) Read Ghilani Chapter 3. Occurrences of Random Errors: 1) To analyze random errors , consider the measurements shown in Table 1. a. Here an angle was measured 100 times, and the observed values and their frequency of occurrence are listed provided. 2) The measured values have a range of __________________________________________. 3) Given these many numbers, it can be challenging to see the distribution in a table. 4) Therefore, to help identify a general trend in the data, the number of occurrences can be binned and plotted in a ________________________ plot or a ______________________ . a. This is shown in Figure 2.

Error Theory © R.L Wood and M.E. Mohammadi, 2021 Page 2 of 7 Table 1. Angle measurements, frequencies, and residual values (courtesy of Ghilani 2018).

Error Theory © R.L Wood and M.E. Mohammadi, 2021 Page 3 of 7 Figure 2. Occurrences or frequencies of residual values (courtesy of Ghilani 2018). 5) A frequency plot can show a general trend in the measured values. 6) However, a _____________________can depict the general trend of the residuals more clearly by grouping values into intervals or bins . a. This is shown in Table 2.

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Error Theory © R.L Wood and M.E. Mohammadi, 2021 Page 4 of 7 Table 2. Frequency of residual values per interval or bin (courtesy of Ghilani 2018). 7) There are two approaches to produce a histogram with a reasonable and appropriate number of bins. a. Option 1: decide on the ______________ of intervals or bins and then calculate the interval sizes. b. Option 2: decide on the _______________________ and then calculate the number of bins. 8) In Table 2, the interval size was set to ___________________, which produced 7 intervals. a. This is shown in Figure 3.

Error Theory © R.L Wood and M.E. Mohammadi, 2021 Page 5 of 7 Figure 3. Histogram of the residual values (courtesy of Ghilani 2018). 9) The number of observations in the selected interval(s) indicates the ___________________ of occurrence . 10) A few observations from this histogram above: a. Most of the calculated residual values are closer to ______________ than to the extreme values of ± 5.95. b. A total of _______________observations (out of 100) are within -0.35 to +0.35. That relates to ________________ of the observations. 11) When the top center points of each _________________ are connected with a straight line, the resulted shape is known as a ______________________________________________ . 12) If more observations are performed and the interval size set to a _____________________ value, the frequency polygon approaches a ___________________________________ . a. This is shown in Figure 4.

Error Theory © R.L Wood and M.E. Mohammadi, 2021 Page 6 of 7 Figure 4. Histogram, frequency polygon, and the smooth curve of the residual values (courtesy of Ghilani 2018). 13) When a smooth curve has a shape of a bell, the computed residual values can be modeled via a _____________________ distribution . 14) In surveying, normal distributions are commonly used to approximate ________________ __________________ . 15) In a normal distribution, the area under the curve between two selected values of the horizontal axis demonstrates the probability of ____________________________ . a. This is shown in Figure 5. Figure 5. The normal distribution curve (courtesy of Ghilani 2018).

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Error Theory © R.L Wood and M.E. Mohammadi, 2021 Page 7 of 7 General Comments on Probability and Random Errors: 1) A few additional observations from a histogram of the residual values: a. Small residual values are more frequent than larger residual values. b. Residual values with positive and negative signs of similar magnitude occur with approximately ___________________ frequency or probability. 2) Therefore, a most probable value from a series of measurements can likely __________ _________________________ random errors . Precision Evaluation: 1) The shape of a normal distribution curve can illustrate the error and precision of the measurements . a. If the measurements resulted in lower error values or high precision, the corresponding normal distribution will be ________________________________. i. This is shown in Figure 6a. b. If the measurements resulted in high error values or low precision, the corresponding normal distribution will be ________________________________. i. This is shown in Figure 6b. (a) (b) Figure 6. Normal distribution curves for measurement errors: (a) ___________ precision and (b) ______________ precision errors (courtesy of Ghilani 2018).

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