FUNDAMENTAL SURVEYING Mathematics and Statistics 2 — Questions and Answers
Question 1: What is the most probable value (MPV) in surveying?
- The largest measured value
- The arithmetic mean of a set of equally weighted observations (Correct answer)
- The median of the observations
- The geometric mean of the observations
Correct answer: The arithmetic mean of a set of equally weighted observations
The MPV is the arithmetic mean, which minimizes the sum of the squares of the residuals under the method of least squares.
Question 2: If a traverse has a linear closure of 0.50 ft over a total distance of 2,500 ft, what is the precision ratio?
- 1/5,000 (Correct answer)
- 1/2,500
- 1/1,000
- 1/500
Correct answer: 1/5,000
Precision ratio = closure error / total distance = 0.50 / 2,500 = 1/5,000.
Question 3: What trigonometric formula is used to compute the area of a triangle given two sides and the included angle?
- Area = (1/2) × b × h
- Area = (1/2) × a × b × sin(C) (Correct answer)
- Area = √[s(s-a)(s-b)(s-c)]
- Area = a² + b² − 2ab cos(C)
Correct answer: Area = (1/2) × a × b × sin(C)
The formula Area = (1/2) × a × b × sin(C) uses two sides and the included angle to compute area.
Question 4: A surveyor measures an angle 6 times and gets a standard deviation of 5″. What is the standard error of the mean?
- 2.04″ (Correct answer)
- 5.00″
- 30.00″
- 0.83″
Correct answer: 2.04″
Standard error of the mean = σ / √n = 5 / √6 ≈ 2.04″.
Question 5: What is the departure (ΔE) of a line with a length of 200 ft and a bearing of S 30° W?
- -100.0 ft (Correct answer)
- -173.2 ft
- +100.0 ft
- +173.2 ft
Correct answer: -100.0 ft
Departure = L × sin(bearing angle) × sign = 200 × sin(30°) = 100 ft, negative because it's westward = -100.0 ft.
Question 6: In least squares adjustment, what quantity is minimized?
- Sum of residuals
- Sum of squared residuals (Correct answer)
- Maximum residual
- Sum of absolute residuals
Correct answer: Sum of squared residuals
Least squares adjustment minimizes the sum of the squares of the residuals (v^T P v).
FUNDAMENTAL SURVEYING Mathematics and Statistics 2
What is the most probable value (MPV) in surveying?