FS Exam Preparation

Comprehensive preparation for the Fundamentals of Surveying (FS) exam. 7 modules covering all 7 exam domains with 60 in-depth topics.

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Lesson 4

Error Analysis & Propagation

Learning Objectives

After completing this topic, you should be able to:

Classify errors as systematic, random, or blunders
Apply the law of propagation of errors for sums, differences, products, and functions
Compute the propagated error in a computed quantity
Explain the difference between absolute and relative error
Apply error propagation to common surveying computations

Overview

Error analysis identifies and classifies the sources of measurement error. Error propagation determines how measurement uncertainties combine to affect computed results. When a surveyor measures multiple quantities and combines them through calculations, the errors in the individual measurements propagate into the final result. Understanding this process is essential for designing surveys to meet accuracy specifications and for certifying that results are within tolerance.

Key Concepts

$Figure FS.7.4 — Error Propagation in Surveying$

Types of Errors

Type	Description	Example	How to Handle
Systematic	Consistent, repeatable errors that follow a pattern	Instrument calibration offset, temperature expansion of a steel tape	Identify and correct (calibrate, apply corrections)
Random	Unpredictable variations in measurements	Reading a scale slightly high or low each time	Cannot eliminate; reduce by repeated measurements
Blunders (gross errors)	Mistakes caused by human error	Reading the wrong number, transposing digits, sighting the wrong target	Detect through redundant measurements and checks; eliminate by remeasuring

Key distinctions:

Systematic errors have a predictable pattern and can be modeled and corrected
Random errors are unpredictable but follow a normal distribution when a large number of measurements are made
Blunders are neither systematic nor random -- they are mistakes that must be detected and removed

Absolute vs. Relative Error

Absolute error = |measured value - true value| (in the same units as the measurement)

Relative error = absolute error / true value (dimensionless, often expressed as a ratio or ppm)

Example: A distance measured as 500.03 ft when the true value is 500.00 ft:

Absolute error = 0.03 ft
Relative error = 0.03 / 500 = 1:16,667 or 60 ppm

The General Law of Error Propagation

If a quantity Q is computed from measured values x1, x2, ..., xn:

Q = f(x1, x2, ..., xn)

The variance of Q is:

sigma_Q^2 = (dQ/dx1)^2 * sigma_x1^2 + (dQ/dx2)^2 * sigma_x2^2 + ... + (dQ/dxn)^2 * sigma_xn^2

Where dQ/dxi is the partial derivative of Q with respect to xi.

This assumes the errors are independent (uncorrelated).

Common Propagation Formulas

Sum or difference: Q = a +/- b

sigma_Q = sqrt(sigma_a^2 + sigma_b^2)

Note: Errors add in quadrature whether the quantities are added or subtracted.

Product: Q = a * b

(sigma_Q / Q)^2 = (sigma_a / a)^2 + (sigma_b / b)^2

Quotient: Q = a / b

(sigma_Q / Q)^2 = (sigma_a / a)^2 + (sigma_b / b)^2

(Same formula as for products.)

Constant multiplier: Q = k * a (where k is a constant)

sigma_Q = k * sigma_a

Power: Q = a^n

sigma_Q / Q = |n| * sigma_a / a

(The relative error is always positive; the absolute value of n applies when n is negative — e.g., Q = 1/a² has the same magnitude rule as Q = a², with σ_Q / Q = 2·σ_a / a.)

Propagation in Surveying Computations

Distance from n equal tape lengths:

If each tape length has standard deviation sigma_tape:

sigma_total = sigma_tape * sqrt(n)

Example: A 100 ft tape with sigma = 0.01 ft is used 10 times to measure a distance:

sigma_total = 0.01 * sqrt(10) = 0.032 ft

Elevation difference from differential leveling:

If each setup has standard deviation sigma_setup:

sigma_total = sigma_setup * sqrt(number of setups)

Horizontal distance from slope distance and vertical angle:

HD = SD * cos(alpha)

sigma_HD = sqrt((cos(alpha) * sigma_SD)^2 + (SD * sin(alpha) * sigma_alpha)^2)

Where sigma_alpha must be in radians.

Area from coordinates:

If each coordinate has standard deviation sigma_coord, the area error depends on the specific geometry. For a rectangle with sides L and W:

sigma_Area = sqrt((W * sigma_L)^2 + (L * sigma_W)^2)

Practical Implications

Errors accumulate as the square root of the number of measurements, not linearly
This means doubling the number of tape lengths only increases the error by a factor of sqrt(2) = 1.414
Errors in angles must be converted to radians before propagation (unless the formula specifically uses degrees)
In a product, the component with the largest relative error dominates the propagated error
In a sum, the component with the largest absolute error dominates

Worked Example

A distance is measured as the sum of two segments:

Segment 1: 250.00 ft, sigma = 0.02 ft
Segment 2: 350.00 ft, sigma = 0.03 ft

Total distance: D = 250.00 + 350.00 = 600.00 ft

Propagated error:

sigma_D = sqrt(0.02^2 + 0.03^2) = sqrt(0.0004 + 0.0009) = sqrt(0.0013) = 0.036 ft

Note: The propagated error (0.036 ft) is LESS than the simple sum of errors (0.05 ft) because errors are unlikely to be at their maximum values simultaneously.

Common wrong path — mixing absolute and relative error formulas. Sums and differences propagate absolute errors in quadrature ( $\sigma_Q = \sqrt{\sigma_a^2 + \sigma_b^2}$ ). Products and quotients propagate relative errors in quadrature ( $\left(\frac{\sigma_Q}{Q}\right)^2 = \left(\frac{\sigma_a}{a}\right)^2 + \left(\frac{\sigma_b}{b}\right)^2$ ). A common exam mistake: plugging absolute errors into the product formula, or plugging relative errors into the sum formula. If the operation is + or −, keep units (feet, meters). If the operation is × or ÷, convert to ratios first (σ/a, σ/b) and apply the formula to ratios, then multiply back by Q at the end. When in doubt, fall back to partial derivatives: $\sigma_Q^2 = \sum (\partial Q/\partial x_i)^2 \sigma_{x_i}^2$ — that form never lies.

Quick retrieval check — try before reading on.

▶A rectangular lot measures L = 125.00 ft (σ = 0.04 ft) by W = 80.00 ft (σ = 0.03 ft). What is the propagated error in the area?

Area $= L \times W = 125 \times 80 = 10{,}000$ sq ft. Apply the product formula to relative errors:

$(σ_L/L)² = (0.04/125)² = (3.2 × 10^{-4})²$ $(σ_W/W)² = (0.03/80)² = (3.75 × 10^{-4})²$

$(σ_A/A)² = (3.2)² × 10^{-8} + (3.75)² × 10^{-8} = (10.24 + 14.0625) × 10^{-8} = 2.43 × 10^{-7}$

$σ_A/A = \sqrt{2.43 × 10^{-7}} = 4.93 × 10^{-4}$ , so $σ_A = 10{,}000 × 4.93 × 10^{-4} = \mathbf{4.93}$ sq ft.

Equivalently, using $σ_A = \sqrt{(W σ_L)^2 + (L σ_W)^2} = \sqrt{(80 × 0.04)^2 + (125 × 0.03)^2} = \sqrt{10.24 + 14.0625} = \sqrt{24.30} = 4.93$ sq ft. Same answer, two routes.

Exam Tips

Errors always add in quadrature (square root of sum of squares), never linearly
For sums and differences, propagate absolute errors; for products and quotients, propagate relative errors
A constant multiplier scales the error linearly: sigma_Q = k * sigma_a
Converting angle error to linear error: linear error = distance * angle_error_in_radians
To convert arcseconds to radians: multiply by (pi / 648000) or approximately by (1 / 206265)
Error propagation problems are among the most commonly tested topics in the math/statistics section
Always check that angle errors are in radians before propagating -- this is the most common mistake
The general formula (partial derivatives) looks complex but simplifies to simple rules for common operations

FS Exam Preparation

Module 1: Surveying Processes & Methods

Module 2: Mapping Processes & Methods

Module 3: Boundary Law & Real Property

Module 4: Surveying Principles & Geodesy

Module 5: Survey Computations

Module 6: Business Concepts

Module 7: Applied Math & Statistics