FS Module 7: Applied Math & Statistics

What is standard deviation (sigma) and how is it calculated?

A measure of the dispersion of measurements about the mean. sigma = sqrt(sum(v_i^2) / (n-1)) where v_i = residual (observation - mean), n = number of observations. For a normal distribution, ~68% of observations fall within +/- 1 sigma of the mean.

What is the probability distribution most commonly assumed for random survey errors?

The normal (Gaussian) distribution. It is symmetric, bell-shaped, centered on the mean, and characterized by mean (mu) and standard deviation (sigma). Most random errors in surveying (pointing, reading, centering) follow this distribution, which is the foundation for least squares adjustment and statistical testing.

What is the weight of a measurement in least squares?

Weight (w) = sigma_0^2 / sigma_i^2, or simply w = 1/sigma_i^2 (when sigma_0 = 1). Weight is inversely proportional to variance. A measurement twice as precise (half the sigma) gets 4 times the weight. For leveling, weight is often proportional to 1/distance (since sigma^2 is proportional to distance).

What is the rule for significant figures in multiplication/division?

The result should have the same number of significant figures as the measurement with the fewest significant figures. Example: 125.3 x 2.1 = 260 (2 sig figs, not 263.13). Leading zeros are not significant; trailing zeros after a decimal are significant.

What are the three types of errors in surveying measurements?

Systematic errors: consistent pattern, same sign and magnitude under same conditions (e.g., incorrect tape length, uncalibrated instrument). Can be corrected. Random errors: unpredictable, follow normal distribution, cannot be eliminated but can be reduced by averaging. Blunders (mistakes): gross errors from carelessness (wrong point, transposed numbers). Detected by checks and redundancy.

What is the standard error of the mean?

sigma_mean = sigma / sqrt(n) where sigma is the standard deviation of individual measurements and n is the number of measurements. The mean is always more precise than any individual measurement. Doubling the number of measurements improves the mean's precision by a factor of sqrt(2), not 2.

What is the rule for significant figures in addition/subtraction?

The result should have the same number of decimal places as the measurement with the fewest decimal places. Example: 125.34 + 2.1 = 127.4 (not 127.44). This differs from multiplication/division, which uses the fewest significant figures.

How do errors propagate for a sum of independent measurements?

If Z = A + B + C (independent measurements), then: sigma_Z = sqrt(sigma_A^2 + sigma_B^2 + sigma_C^2) Variances always ADD, whether the quantities are added or subtracted. Example: if each of 5 level setups has sigma = +/-0.005 ft, the total sigma = sqrt(5 x 0.005^2) = +/-0.011 ft.

What are degrees of freedom and why do they matter?

Degrees of freedom = number of observations minus number of unknowns (n - u). Also called redundancy. In a traverse with n measured angles and u unknown coordinates, the degrees of freedom represent the number of independent checks. More degrees of freedom = better statistical reliability. Zero degrees of freedom = no checks on data quality.

What is the probable error (PE)?

PE = 0.6745 x sigma. It defines a range within which 50% of observations are expected to fall. It was historically common in surveying but has largely been replaced by standard deviation. Sometimes seen in older specifications. PE50 = 0.6745 sigma, PE90 = 1.6449 sigma, PE95 = 1.96 sigma.

How does leveling precision relate to distance?

For differential leveling, the standard deviation of the elevation difference is proportional to the square root of the distance: sigma = c x sqrt(D), where c is a constant depending on equipment/procedure and D is the one-way distance in km. This is why leveling specifications are stated as, e.g., "12 mm x sqrt(K)" where K is the distance in km.

What are the confidence intervals for a normal distribution?

+/- 1 sigma: 68.27% of observations +/- 2 sigma: 95.45% of observations +/- 3 sigma: 99.73% of observations +/- 1.96 sigma: exactly 95% +/- 2.576 sigma: exactly 99% For the FS exam, remember: 68-95-99.7 rule.

What is a weighted mean and when is it used?

Weighted mean = sum(w_i x x_i) / sum(w_i) Used when measurements have different precisions. The weight of a measurement is inversely proportional to its variance: w = 1/sigma^2 (or proportional to the number of measurements if averaging sets). More precise measurements get higher weight.

What is the most probable value (MPV) of a set of equally weighted measurements?

The arithmetic mean. When all measurements are equally precise (equally weighted), the mean minimizes the sum of squared residuals, making it the most probable value under the principle of least squares. For unequally weighted measurements, the weighted mean is the MPV.

How do you propagate error through a function of one variable?

If Z = f(A), then sigma_Z = |dZ/dA| x sigma_A The error in Z equals the absolute value of the derivative times the error in A. Example: if Area = L^2 and sigma_L = 0.05 ft, then sigma_Area = |2L| x 0.05 ft. For L = 100 ft: sigma_Area = 200 x 0.05 = 10 sq ft.

What is the general law of error propagation?

For Z = f(A, B, C, ...): sigma_Z^2 = (dZ/dA)^2 x sigma_A^2 + (dZ/dB)^2 x sigma_B^2 + ... + 2(dZ/dA)(dZ/dB) x sigma_AB + ... When measurements are independent, the covariance terms (sigma_AB) are zero, simplifying to just the sum of squared partial derivatives times variances.

What is the 95% confidence interval for a measurement?

x_bar +/- t_(alpha/2,n-1) x sigma_mean For large samples (n > 30), t approaches 1.96, so the 95% CI is approximately x_bar +/- 1.96 x sigma/sqrt(n). For small samples, use the t-distribution with (n-1) degrees of freedom. The confidence interval means there is a 95% probability the true value lies within this range.