Types of Errors

Overview#

No measurement is exact. This is the foundational premise of error theory and the starting point for all rigorous surveying practice. Every time a surveyor reads an angle, measures a distance, or determines an elevation, the recorded value differs from the true value by some amount. That difference is the measurement error.

The true value of any quantity is never known. If it were, there would be no need to measure. Instead, surveyors work with the most probable value -- the best estimate of the true value derived from repeated measurements and statistical analysis. The quality of a survey is not measured by achieving zero error (an impossibility) but by understanding, quantifying, and controlling the errors present in the work.

"Since the true value of a measurement is never known, the exact error present is always unknown." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 2, p. 28

Errors in surveying measurements are classified into three fundamental categories: systematic errors, random errors, and blunders (mistakes). Each type has distinct characteristics, distinct causes, and distinct methods of treatment. Understanding these distinctions is the first step toward managing measurement quality and producing results that meet professional standards.

The mathematical treatment of errors -- standard deviations, confidence intervals, error propagation, least squares adjustment -- all operates on the assumption that systematic errors have been removed and blunders have been eliminated, leaving only random errors for statistical analysis. If that assumption is violated, no amount of mathematical sophistication will save the survey.

Systematic Errors#

Systematic errors follow a fixed pattern or mathematical law. Under the same measurement conditions, a systematic error will have the same magnitude and the same sign every time. This predictability is both their defining characteristic and the key to eliminating them.

Characteristics

• Constant or predictable under repeated conditions
• Cumulative -- they do not cancel with repeated measurements; they accumulate
• Deterministic -- they can be modeled, computed, and removed
• Directional -- they bias results consistently in one direction (too long, too short, too high, too low)

Because systematic errors do not change sign randomly, taking the mean of many measurements does not reduce them. A steel tape that is 0.01 ft too long will measure every 100-ft distance 0.01 ft too short, whether the surveyor measures it once or a hundred times. The average of 100 such measurements is still 0.01 ft too short.

Common Sources in Surveying

Elimination

Systematic errors are handled by:

1. Calibration. Comparing instruments against known standards and applying correction factors. A steel tape is standardized against a baseline; an EDM is checked against a calibration baseline; a level is tested for collimation error.
2. Applying corrections. Temperature corrections, pressure corrections, curvature and refraction corrections, sea-level corrections, grid scale factors -- these are all mathematical models that remove known systematic effects.
3. Proper field procedures. Balancing backsight and foresight distances in leveling eliminates collimation error. Measuring angles in both direct and reverse (face left and face right) cancels several instrumental systematic errors. These procedures are not optional niceties; they are systematic-error defenses built into standard practice.

"Systematic errors can be computed and corrections can be applied to the measurements to eliminate these errors. Or, procedures can be adopted in the field that will automatically eliminate them." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 2, p. 29

If systematic errors are not corrected, they propagate through every computation that depends on the affected measurements. A survey closed with an apparently good precision ratio may still be positionally inaccurate if pervaded by uncorrected systematic error -- the error simply accumulates consistently, creating an internal geometry that is precise but biased.

Random Errors#

Random errors are the errors that remain after all systematic errors have been corrected and all blunders have been eliminated. They are caused by the inherent limitations of instruments, observers, and environmental conditions -- factors that vary unpredictably from one measurement to the next.

Characteristics

Random errors obey the laws of probability and follow a normal (Gaussian) distribution. This statistical behavior is described by four key properties:

1. Small errors occur more frequently than large ones. The probability density is greatest near zero error and decreases symmetrically outward.
2. Positive and negative errors of the same magnitude are equally likely. The distribution is symmetric about zero. Over many measurements, positive and negative random errors tend to cancel.
3. Very large errors are extremely rare. The tails of the normal distribution approach but never reach zero, meaning a very large random error is possible but has vanishingly small probability.
4. The mean of the errors approaches zero as the number of measurements increases. This is the practical consequence of properties 1--3 and the theoretical basis for improving precision through repeated measurement.

The normal distribution of random errors is described by the probability density function:

$f(x) = \frac{1}{\sigma\sqrt{2\pi}} \, e^{-\frac{(x - \mu)^2}{2\sigma^2}}$

where $\mu$ is the population mean and $\sigma$ is the population standard deviation. For measurement errors, $\mu$ is assumed to be zero (after systematic errors are removed), and $\sigma$ characterizes the spread -- the precision -- of the measurements.

Why Random Errors Matter

Random errors are the subject of measurement statistics. Standard deviation, probable error, confidence intervals, error propagation, and least squares adjustment all deal specifically with random errors. The entire mathematical apparatus of survey adjustment assumes that the errors being adjusted are random.

"The mathematical theory of probability can be applied to random errors, making it possible to evaluate their effects on computed survey quantities." -- Ghilani, Adjustment Computations (6th Ed.), Ch. 2, p. 18

Taking the mean of repeated measurements reduces the effect of random errors. If a single measurement has standard deviation $\sigma$, the standard deviation of the mean of $n$ independent measurements is:

$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

This is the mathematical justification for redundant measurement. Four measurements reduce the standard deviation of the mean by half; to reduce it by half again requires sixteen. The law of diminishing returns is built into the $\sqrt{n}$ relationship, which is why practical surveying balances the number of repeated measurements against the time and cost of making them.

Blunders (Mistakes)#

Blunders are not errors in the mathematical or statistical sense. They are mistakes -- gross, typically large deviations caused by human carelessness, misunderstanding, inattention, or equipment malfunction. No probability distribution describes them. No adjustment method corrects them. They must be detected and eliminated before any statistical analysis is meaningful.

Common Blunders in Surveying

• Misreading a rod or scale -- reading 5.83 ft instead of 6.83 ft (one-foot error)
• Transposing digits -- recording 285.47 as 258.47
• Setting up on the wrong point -- occupying a point other than the intended survey station
• Sighting the wrong target -- turning an angle to the wrong prism or reflector
• Incorrect instrument height -- measuring or recording the HI (height of instrument) incorrectly
• Arithmetic errors -- mistakes in field reduction, summing, or transcription
• Using wrong units -- entering meters when the software expects feet, or vice versa
• Taping blunders -- losing a full tape length in a long measurement, or misidentifying an intermediate point

Detection Methods

Because blunders follow no mathematical pattern, they cannot be modeled or corrected -- only detected and removed. The primary defense against blunders is redundancy:

1. Redundant measurements. Measuring the same quantity more than the minimum number of times needed. If three angles in a triangle should sum to $180°$ and they sum to $193°$, a blunder is present.
2. Closure checks. Comparing the sum of parts to a known whole. Level loop misclosures, traverse angular closures, and distance checks all reveal blunders by comparing measured results against geometric constraints.
3. Independent verification. Having a second observer check readings, or measuring the same quantity by a different method.
4. Statistical outlier detection. After adjustment, examining residuals for values that are statistically improbable. A residual that exceeds $3\sigma$ has a probability of only 0.3% under the normal distribution -- a strong indicator that the measurement is a blunder, not a random error.

"Mistakes are caused by the carelessness of the observer. They are not errors in the true sense, since they usually arise from the observer's inattention or misunderstanding, and they do not follow any mathematical law." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 2, p. 28

The single most important principle in blunder prevention is that the observer should never assume a measurement is correct without a check. Every field measurement should have some form of independent verification built into the procedure.

Sources of Error in Surveying#

Errors arise from three broad categories of sources. Understanding these categories helps surveyors anticipate potential problems and design procedures that minimize their effects.

Instrumental Errors

These arise from imperfections in the equipment itself: a level that is out of adjustment, a tape that is not standard length, an EDM with an incorrect calibration constant, a total station with an uncorrected index error. Regular calibration and proper field checks (e.g., two-peg test for levels, face left/face right for total stations) are the primary defenses.

Natural Errors

These arise from environmental conditions that affect the measurement: temperature changes expanding or contracting a steel tape, atmospheric refraction bending the line of sight, wind vibrating a plumb bob, humidity affecting EDM signal propagation, gravity variations affecting level surfaces. Natural errors can be systematic (curvature and refraction) or random (thermal turbulence).

Personal Errors

These arise from human limitations: the observer's ability to center a bubble, to bisect a target, to estimate between the smallest divisions of a scale, to hold a tape at the correct tension. Personal errors include both random components (slight variations in pointing and reading) and systematic components (personal bias in interpolating readings). Blunders are also personal in origin, though they are treated as a separate category because of their gross nature.

Error Sources by Survey Operation

Error vs. Residual#

Two concepts that are often confused but are fundamentally different: the true error and the residual.

True Error

The true error of a measurement is the difference between the measured value and the true value:

$\epsilon = M - T$

where $\epsilon$ is the true error, $M$ is the measured value, and $T$ is the true value. Since the true value is never known, the true error is unknowable. It exists as a concept but can never be computed for an actual measurement.

Residual

The residual is the difference between the measured value and the most probable value:

$v = M - \hat{X}$

where $v$ is the residual, $M$ is the measured value, and $\hat{X}$ is the most probable value (typically the mean of repeated measurements, or the adjusted value from a least squares adjustment).

Residuals are computable. They are what surveyors actually work with when evaluating measurement quality, computing standard deviations, and performing adjustments. The residual is an estimate of the true error, and for a well-designed survey with sufficient redundancy, it is a reliable one.

The Most Probable Value

For a set of equally weighted measurements of the same quantity, the most probable value is the arithmetic mean:

$\hat{X} = \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$

This result follows directly from the assumption that random errors follow a normal distribution. The mean minimizes the sum of the squares of the residuals -- a property that connects it to the principle of least squares.

"The most probable value of a quantity that has been measured several times under the same conditions is the arithmetic mean of the observations." -- Ghilani, Adjustment Computations (6th Ed.), Ch. 2, p. 17

When measurements have unequal weights (because they were made under different conditions or with different instruments), the most probable value is the weighted mean, where each measurement is weighted in proportion to its reliability.

Eliminating and Minimizing Errors#

Each type of error requires a different strategy. Effective surveying practice deploys all three strategies simultaneously.

Systematic Errors: Calibration and Correction

• Calibrate instruments regularly against known standards
• Apply mathematical corrections for known physical effects (temperature, pressure, curvature, refraction)
• Use field procedures designed to cancel systematic effects (balanced sight distances, direct/reverse measurements)
• Verify corrections by checking against independent measurements

Blunders: Redundancy and Checking

• Build redundancy into every measurement scheme -- never rely on a single, unchecked measurement
• Use closure checks (angular, distance, elevation) to catch gross errors immediately in the field
• Maintain clear, organized field notes to reduce transcription errors
• Implement independent checks by different observers or methods
• Review data before leaving the field site

Random Errors: Statistical Analysis and Adjustment

• Take repeated measurements to improve the estimate of the most probable value
• Use least squares adjustment to distribute random errors optimally across a measurement network
• Compute and report standard deviations and confidence intervals to quantify uncertainty
• Compare achieved precision against project specifications and professional standards

The Goal

The goal is not zero error -- that is impossible. The goal is quantified, acceptable error: error that has been identified by type, reduced by appropriate methods, and reported with honest statistical measures so that the users of the survey know exactly how much confidence to place in the results.

"The knowledge of precision and accuracy of surveying measurements is essential for properly designing, executing, and evaluating surveys." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 2, p. 27

A professional surveyor who understands error theory does not merely produce numbers. They produce numbers accompanied by a statement of their quality -- and they can defend that statement with evidence.

Key Takeaways#

• Every measurement contains error. The true value is never known; surveyors work with the most probable value derived from redundant measurements and statistical analysis.
• Systematic errors are predictable and cumulative. They follow mathematical laws and can be removed through calibration, corrections, and proper field procedures. They do not cancel with repeated measurements.
• Random errors remain after systematic errors are removed. They follow the normal distribution, are equally likely to be positive or negative, and are reduced (but never eliminated) by averaging repeated measurements. Random errors are what statistics and least squares adjustment address.
• Blunders are gross mistakes caused by human carelessness or equipment malfunction. They follow no mathematical law and must be detected and removed through redundancy, closure checks, and outlier analysis. No adjustment can fix a blunder -- only eliminate it.
• Sources of error are instrumental, natural, and personal. Effective survey design addresses all three through calibration, environmental corrections, and careful field procedures.
• Residuals (computable) are the practical stand-in for true errors (unknowable). The most probable value of equally weighted measurements is the arithmetic mean.
• The goal is not zero error but quantified, acceptable error -- measured, reported, and defensible.

References#

1. Ghilani, C.D. & Wolf, P.R. Elementary Surveying: An Introduction to Geomatics (13th Ed.). Pearson, 2012. Chapter 2: "Theory of Errors in Observations."
2. Ghilani, C.D. Adjustment Computations: Spatial Data Analysis (6th Ed.). Wiley, 2017. Chapters 2--3.
3. Mikhail, E.M. & Gracie, G. Analysis and Adjustment of Survey Measurements. Van Nostrand Reinhold, 1981.
4. U.S. Department of the Interior, Bureau of Land Management. Manual of Surveying Instructions (2009).
5. Federal Geodetic Control Committee. Standards and Specifications for Geodetic Control Networks. NOAA, 1984.