Precision & Accuracy

Overview#

Precision and accuracy are the two pillars of measurement quality, and every practicing surveyor must understand both -- not as interchangeable synonyms, but as distinct concepts that answer fundamentally different questions about a set of measurements.

Accuracy asks: How close is the measurement to the true value? Precision asks: How close are the repeated measurements to each other?

A survey can be highly precise yet grossly inaccurate. It can also be accurate on average yet imprecise. Understanding the distinction allows surveyors to diagnose the source of measurement problems, select appropriate equipment and procedures, meet project specifications, and defend their work in professional and legal contexts.

"Accuracy denotes the absolute nearness of a measured quantity to its true value, whereas precision refers to the degree of refinement or consistency of a group of measurements and their nearness to each other." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (13th Ed.), Ch. 3, p. 41

This distinction is not merely academic. When a surveyor reports a boundary position, a client and reviewing agency need to know both whether the measurements are internally consistent (precision) and whether the reported position is close to reality (accuracy). The Federal Geodetic Control Subcommittee (FGCS), the American Society for Photogrammetry and Remote Sensing (ASPRS), and professional licensing boards all define standards that reference these concepts explicitly.

Accuracy#

Accuracy describes how close a measured value is to the true value (or the best available accepted value) of the quantity being measured. It is the ultimate measure of correctness.

Assessing Accuracy

The challenge with accuracy is that the true value of a measured quantity is never known exactly. In practice, surveyors assess accuracy by comparing their measurements against a value of higher order -- a value determined by methods known to be significantly more precise and carefully controlled. For example, a traverse closure might be compared to NGS-published coordinates for the control points, or an EDM distance might be checked against a calibrated baseline.

The error (or residual in the context of adjustments) of a single measurement is:

$e = x_{\text{measured}} - x_{\text{true}}$

Because the true value is unknown, the accepted or most probable value is used instead:

$v = x_{\text{measured}} - x_{\text{most probable}}$

Sources of Error Affecting Accuracy

Accuracy is degraded by all three categories of error:

• Systematic errors -- Errors that follow a predictable pattern and accumulate in one direction. An uncalibrated EDM, an index error in a total station, or failure to apply curvature and refraction corrections all introduce systematic bias. These errors shift the measured value away from the true value consistently.
• Random errors -- Small, unavoidable fluctuations that follow the laws of probability. They affect both accuracy and precision, though their mean effect on accuracy diminishes as the number of measurements increases.
• Blunders (mistakes) -- Gross errors caused by human mistake, equipment malfunction, or procedural failure. A transposed digit in a field note, sighting on the wrong target, or misreading a rod are blunders that destroy accuracy entirely.

"Systematic errors are caused by factors that act in the same manner on observations... They can be computed and eliminated by applying corrections." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (13th Ed.), Ch. 3, p. 38

Expressing Accuracy

Accuracy can be expressed in several ways:

A ratio of $1:10{,}000$ means one unit of error per ten thousand units measured -- for a 5,000 ft traverse, the allowable misclosure would be $5{,}000 / 10{,}000 = 0.5$ ft.

Precision#

Precision describes the internal consistency of a set of measurements -- how tightly grouped the repeated observations are, regardless of whether they cluster near the true value. A precise survey produces nearly the same result each time the measurement is repeated under the same conditions.

Characteristics of Precision

• Precision is affected primarily by random errors. Systematic errors do not reduce precision; they shift the entire cluster of measurements away from the true value while leaving the internal spread unchanged.
• Precision is a necessary but not sufficient condition for accuracy. A survey must be precise to be accurate, but precision alone does not guarantee accuracy.
• Precision can be evaluated from the measurements themselves, without any knowledge of the true value. This makes precision the more readily available quality metric during fieldwork.

Statistical Measures of Precision

The most common statistical measures of precision are:

Standard Deviation ($\sigma$ or $s$): The most widely used measure of spread. For a sample of $n$ measurements with mean $\bar{x}$:

$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}}$

The denominator uses $n - 1$ (degrees of freedom) rather than $n$ because the sample mean is used instead of the true population mean.

Variance ($s^2$): The square of the standard deviation. Variance is additive for independent measurements, making it essential for error propagation calculations.

$s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}$

Standard Deviation of the Mean ($s_{\bar{x}}$): The precision of the mean of $n$ measurements improves with additional observations:

$s_{\bar{x}} = \frac{s}{\sqrt{n}}$

This relationship explains why surveyors take multiple measurements -- each additional observation improves the precision of the reported value, though with diminishing returns.

"The standard deviation is the most frequently used measure of precision. A small standard deviation indicates high precision, meaning the observations are closely clustered about the mean." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (13th Ed.), Ch. 3, p. 48

90%, 95%, and 99% Confidence Intervals: For normally distributed errors, the probability that a single measurement falls within a given range of the mean can be expressed using multipliers of the standard deviation:

The 95% confidence level has become the standard reporting convention in modern surveying and geospatial practice.

The Target Analogy#

The classic way to illustrate the distinction between precision and accuracy is the target (or dartboard) analogy. Consider four scenarios where a surveyor takes multiple shots at a target whose bullseye represents the true value:

Four Quadrants of Measurement Quality

High Accuracy, High Precision -- The shots form a tight cluster centered on the bullseye. Repeated measurements are consistent with each other (precise) and their mean coincides with the true value (accurate). This is the goal of every survey. Systematic errors have been eliminated; random errors are small.

High Accuracy, Low Precision -- The shots are scattered widely, but their center (mean) falls on the bullseye. The individual measurements vary considerably, but the average is correct. This situation is uncommon in practice because the wide scatter itself usually indicates a problem.

Low Accuracy, High Precision -- The shots form a tight cluster, but the cluster is displaced from the bullseye. The measurements are internally consistent, yet they are all wrong by roughly the same amount. This is the hallmark of an undetected systematic error -- perhaps an uncalibrated instrument, an incorrect prism constant, or a datum transformation error. The survey "looks good" internally but is offset from truth.

Low Accuracy, Low Precision -- The shots are scattered widely and their center does not coincide with the bullseye. Both systematic and random errors are large. This indicates fundamental problems with equipment, procedures, or field conditions.

"Precise measurements may not necessarily be accurate, nor must accurate measurements necessarily be precise. The distinction between precision and accuracy is illustrated by the rifle shooting example." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (13th Ed.), Ch. 3, p. 42

The critical lesson for practicing surveyors: precision is observable in the field; accuracy often is not. A tight traverse closure (good precision) feels reassuring, but if every angle was turned with the same instrument error, the traverse may close beautifully while being systematically displaced from its true position. Independent checks -- ties to higher-order control -- are essential to verify accuracy.

Relative vs. Absolute Accuracy#

Modern surveying distinguishes between two fundamentally different ways of expressing accuracy, and confusing them is a common source of misunderstanding.

Relative Accuracy

Relative accuracy describes the accuracy of the spatial relationship between two points -- the distance and direction between them -- without reference to their absolute positions in a coordinate system.

It is typically expressed as a ratio:

$\text{Relative Accuracy} = \frac{\text{error in the distance between two points}}{\text{distance between the two points}}$

For example, a relative accuracy of $1:20{,}000$ means that for every 20,000 units of distance measured, the error is no more than 1 unit. Over a 10,000 ft line, the expected error would be no more than:

$\frac{10{,}000}{20{,}000} = 0.5 \text{ ft}$

Relative accuracy is the traditional metric for traverse work and is still used in many state standards for boundary surveys.

Absolute Accuracy (Positional Accuracy)

Absolute accuracy (also called positional accuracy or network accuracy) describes how well a point's coordinates match its true position within a defined reference frame (e.g., NAD 83). It is expressed as a positional uncertainty, typically at the 95% confidence level:

\text{Positional Accuracy} = \pm \, \text{value (in meters) at 95% confidence}

For example, a point with a positional accuracy of $\pm 0.02$ m at 95% confidence means there is a 95% probability that the true position falls within a circle of 0.02 m radius centered on the published coordinates.

Network Accuracy vs. Local Accuracy

The National Geodetic Survey (NGS) further distinguishes:

• Network accuracy -- The accuracy of a point's coordinates relative to the National Spatial Reference System (NSRS) datum. This is an absolute measure.
• Local accuracy -- The accuracy of a point's coordinates relative to other directly connected, adjacent points. This is closer to relative accuracy.

A control point can have excellent local accuracy (its position relative to neighboring stations is very well determined) while having poorer network accuracy (its absolute position within the NSRS carries greater uncertainty). Both metrics matter, but for different purposes. Boundary surveys often care most about local accuracy; geodetic networks prioritize network accuracy.

Standards of Accuracy#

The Federal Geodetic Control Subcommittee (FGCS) has established classification standards for horizontal and vertical control surveys. These standards define the minimum accuracy required for different orders of survey work.

Horizontal Control Standards

These standards apply to the relative accuracy between directly connected adjacent survey stations. Higher-order surveys require more rigorous procedures, better equipment, and more redundant observations.

Vertical Control Standards

Vertical accuracy standards are expressed using the formula:

$\text{Maximum allowable discrepancy} = c\sqrt{K}$

where $K$ is the distance in kilometers between benchmarks along the leveling route, and $c$ is a constant that depends on the order and class:

"The FGCS has established standards of accuracy and specifications for geodetic surveys which are used by all federal mapping and surveying agencies." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (13th Ed.), Ch. 19, p. 561

First-order control forms the backbone of the national geodetic network. Most boundary and construction surveys operate at the Second or Third Order level. The order required depends on the project: a statewide geodetic densification might demand First Order; a residential lot survey might need only Third Order, Class II.

Positional Accuracy Standards#

As GPS and GNSS technology transformed surveying, traditional relative accuracy standards proved insufficient to describe the quality of satellite-derived positions. Modern standards have shifted toward positional accuracy metrics.

NSRS Positional Accuracy

The National Geodetic Survey reports the accuracy of points in the National Spatial Reference System using positional accuracy values at the 95% confidence level. For horizontal positions, this is computed as:

$A_{95} = 2.4477 \times \text{RMSE}_r$

where $\text{RMSE}_r$ is the root mean square error of the radial (horizontal) position. The multiplier $2.4477$ converts from RMSE to the 95% confidence circle, assuming equal and independent errors in the northing and easting components.

For vertical positions:

$A_{95} = 1.960 \times \text{RMSE}_z$

ASPRS Positional Accuracy Standards

The American Society for Photogrammetry and Remote Sensing (ASPRS) publishes positional accuracy standards for geospatial data, including surveyed points and mapped features. The key concepts include:

• RMSE (Root Mean Square Error) -- The square root of the average of the squared differences between measured and true positions. This is the fundamental accuracy metric:

$\text{RMSE}_x = \sqrt{\frac{\sum_{i=1}^{n}(x_i - x_{\text{true},i})^2}{n}}$

• Circular Error (Horizontal) -- For 2D horizontal accuracy, RMSE values in $x$ and $y$ are combined:

$\text{RMSE}_r = \sqrt{\text{RMSE}_x^2 + \text{RMSE}_y^2}$

• 95% Confidence Reporting -- The convention across federal agencies, professional societies, and most state standards is to report accuracy at the 95% confidence level. This means there is a 95% probability that the true position lies within the stated uncertainty of the reported position.

CEP vs. RMSE

Circular Error Probable (CEP) is the radius of a circle centered on the true position that contains 50% of the measured positions. CEP is commonly used in military and navigation applications but less so in surveying. The relationship between CEP and RMSE for circular normal distributions is approximately:

$\text{CEP} \approx 0.589 \times (\sigma_x + \sigma_y)$

Surveying practice favors RMSE and the 95% confidence circle because they provide a more conservative (and more useful) characterization of positional uncertainty.

Specifications vs. Standards#

Surveyors frequently encounter the terms "standards" and "specifications" used together, and it is important to understand the distinction.

Standards

Standards define the required accuracy for a given classification of survey. They answer the question: What level of accuracy must the survey achieve? Standards are expressed as accuracy values -- ratios, positional uncertainties, or closure limits. The FGCS accuracy classifications discussed above are standards.

Standards are outcome-based. They tell you the result you must achieve but not how to achieve it.

Specifications

Specifications describe the procedures, equipment, methods, and field practices required to achieve a given standard. They answer the question: How must the survey be performed to meet the standard?

Specifications might dictate:

• Minimum number of sets of angles
• Maximum allowable distance between instrument setups
• Required instrument calibration frequency
• Observation scheduling (e.g., daylight-only for precise leveling)
• Redundancy requirements (e.g., minimum number of ties to existing control)
• Equipment requirements (e.g., minimum instrument precision of $\pm 1''$ for First Order angles)

"Standards specify the accuracy to be attained, and specifications outline the procedures necessary to achieve the accuracy." -- Ghilani & Wolf, Elementary Surveying: An Introduction to Geomatics (13th Ed.), Ch. 19, p. 562

A surveyor who follows all specifications but fails to meet the accuracy standard has not completed the survey successfully. Conversely, meeting the accuracy standard through non-standard procedures may raise questions about the reliability and legal defensibility of the work. Professional practice requires adherence to both.

Practical Application

In practice, project requirements usually reference a standard (e.g., "Third Order, Class I horizontal control"), and the surveyor selects equipment and field procedures that meet the corresponding specifications. The surveyor then verifies through statistical analysis of the observations -- closures, residuals, chi-square testing -- that the standard has been achieved.

Key Takeaways#

• Accuracy and precision are different. Accuracy measures closeness to the true value; precision measures consistency among repeated measurements. Both must be evaluated to fully characterize measurement quality.
• Precision is necessary but not sufficient for accuracy. A systematic error can produce a highly precise yet grossly inaccurate survey. Independent checks against higher-order control are essential.
• Systematic errors affect accuracy; random errors affect both. Eliminating systematic errors through calibration, proper procedures, and corrections is the primary path to accuracy. Reducing random errors through repeated observations and better equipment improves precision.
• Relative and absolute accuracy serve different purposes. Relative accuracy describes the quality of measurements between points; absolute (positional) accuracy describes a point's position within a reference frame.
• Modern standards use positional accuracy at 95% confidence. The profession has moved from simple closure ratios toward RMSE-based positional accuracy reporting, reflecting the shift to GNSS-era surveying.
• Standards define the goal; specifications define the path. Both are required for professional survey work.
• Always verify accuracy independently. Good internal precision can mask systematic errors. Tie to higher-order control. Compare against independent measurements. Trust the math, but check the work.

References#

• Ghilani, C. D. & Wolf, P. R. (2012). Elementary Surveying: An Introduction to Geomatics (13th Ed.). Pearson.
• Federal Geodetic Control Subcommittee (FGCS). (1984). Standards and Specifications for Geodetic Control Networks. National Oceanic and Atmospheric Administration.
• National Geodetic Survey (NGS). Guidelines for Establishing GPS-Derived Ellipsoid Heights. NOAA Technical Memorandum NOS NGS-58.
• American Society for Photogrammetry and Remote Sensing (ASPRS). (2015). ASPRS Positional Accuracy Standards for Digital Geospatial Data.
• National Geodetic Survey. National Spatial Reference System (NSRS) Network Accuracy and Local Accuracy Definitions.