Fundamentals of Surveying Exam (FS) Mathematics, Statistics, and Geodesy Syllabus

Write the formula for the sample standard deviation $s$ of $n$ measurements (with $n-1$ in the denominator).

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

What is the difference between variance and standard deviation?

Variance is the mean of squared deviations ($s^{2}$); standard deviation is its square root ($s$), expressed in the same units as the data.

Define the standard deviation (standard error) of the mean $s_{\bar{x}}$ and how it changes with the number of observations.

$$s_{\bar{x}} = \frac{s}{\sqrt{n}}$$ It decreases as $n$ increases, i.e., averaging more observations improves precision proportionally to $\sqrt{n}$.

In surveying, what is the 'most probable value' (MPV) of a quantity measured several times with equal weight?

The MPV is the arithmetic mean of the observations, $\bar{x} = \dfrac{\sum x_i}{n}$, which minimizes the sum of squared residuals (least squares for a single quantity).

What is the relationship between standard error ($\sigma$, 68.3%), the 90% error ($E_{90}$), and the 95% error ($E_{95}$) for a normal distribution?

Approximately $E_{50} = 0.6745\sigma$, $E_{90} = 1.6449\sigma$, $E_{95} = 1.9599\sigma$ (about $1.96\sigma$), and $E_{99.7} = 3\sigma$.

For independent events $A$ and $B$, what is $P(A \text{ and } B)$, and for mutually exclusive events what is $P(A \text{ or } B)$?

Independent (multiplication): $P(A \cap B) = P(A)\,P(B)$. Mutually exclusive (addition): $P(A \cup B) = P(A) + P(B)$.

What is the general addition rule for the probability of $A$ or $B$ when events are not mutually exclusive?

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

What distribution describes random surveying errors, and what are its key shape properties?

The normal (Gaussian) distribution — symmetric, bell-shaped, centered on the true value; small errors are more frequent than large ones, and positive and negative errors are equally likely.

State the empirical (68–95–99.7) rule for a normal distribution.

About $68.3\%$ of values lie within $\pm 1\sigma$, $95.4\%$ within $\pm 2\sigma$, and $99.7\%$ within $\pm 3\sigma$ of the mean.

Name the three classifications of surveying errors and give a one-line definition of each.

Blunders (mistakes): gross human errors, must be eliminated. Systematic errors: follow physical laws, same sign/magnitude under same conditions, can be modeled/corrected. Random (accidental) errors: small, unpredictable in sign, follow probability laws, reduced by averaging.

How do systematic errors and random errors differ in how they accumulate and can be treated?

Systematic errors are cumulative and one-directional; they are removed by calibration or mathematical correction. Random errors are compensating (tend to cancel) and are treated statistically (least squares), never fully eliminated.

Distinguish 'accuracy' from 'precision' in measurement.

Accuracy is closeness of a measurement to the true value (freedom from bias/systematic error). Precision is the closeness/repeatability of measurements to one another (small random scatter).

Give an example of a systematic error in taping (distance measurement) and how it is corrected.

Temperature, tension, sag, or incorrect tape length cause systematic taping errors; corrected with formulas, e.g., temperature correction $C_T = \alpha (T - T_0) L$, sag correction, and standardization (length) correction.

State the law of propagation for the error of a sum (or difference) of independent measurements $x_1, x_2, \dots, x_n$ each with standard deviation $\sigma_i$.

$$\sigma_{\text{sum}} = \sqrt{\sigma_1^{2} + \sigma_2^{2} + \cdots + \sigma_n^{2}}$$

For a series of $n$ measurements each with the same random error $\sigma$, what is the error of the total (series) and the error of the mean?

Error of the series (sum): $E_{\text{series}} = \sigma\sqrt{n}$. Error of the mean: $E_{\text{mean}} = \dfrac{\sigma}{\sqrt{n}}$.

State the general law of error propagation for a function $y = f(x_1, x_2, \dots, x_n)$ of independent variables.

$$\sigma_y = \sqrt{\left(\frac{\partial f}{\partial x_1}\sigma_{x_1}\right)^{2} + \left(\frac{\partial f}{\partial x_2}\sigma_{x_2}\right)^{2} + \cdots + \left(\frac{\partial f}{\partial x_n}\sigma_{x_n}\right)^{2}}$$

For a quantity multiplied by a constant, $y = kx$, how does the error propagate?

$$\sigma_y = |k|\,\sigma_x$$ The error scales directly by the constant.

In weighted observations, how is weight $w$ related to the standard deviation $\sigma$ of a measurement?

Weight is inversely proportional to the variance: $w \propto \dfrac{1}{\sigma^{2}}$. More precise (smaller $\sigma$) observations receive greater weight.

How is the weighted mean computed from observations $x_i$ with weights $w_i$?

$$\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$$

What is the 'figure of the Earth' best approximated by for geodetic computations, and how does it differ from a sphere?

An oblate ellipsoid of revolution (spheroid): flattened at the poles and bulging at the equator, so the equatorial (semi-major) axis $a$ exceeds the polar (semi-minor) axis $b$.

Define the flattening $f$ of a reference ellipsoid in terms of the semi-major axis $a$ and semi-minor axis $b$.

$$f = \frac{a - b}{a}$$ For GRS80/WGS84, $f \approx \dfrac{1}{298.257}$.

Define first eccentricity $e$ of an ellipsoid in terms of $a$ and $b$ (or flattening $f$).

$$e^{2} = \frac{a^{2} - b^{2}}{a^{2}} = 2f - f^{2}$$

What is the geoid, and how does it relate to mean sea level and orthometric heights?

The geoid is the equipotential gravity surface that best approximates global mean sea level, extended under continents. It is the reference for orthometric (elevation) heights and is everywhere perpendicular to the direction of gravity (the plumb line).

State the relationship among ellipsoid height $h$, orthometric height $H$, and geoid height (undulation) $N$.

$$h = H + N$$ where $h$ is height above the ellipsoid, $H$ is height above the geoid (orthometric), and $N$ is the geoid undulation.

Distinguish a horizontal datum from a vertical datum, giving a U.S. example of each.

A horizontal datum defines latitude/longitude positions on a reference ellipsoid (e.g., NAD83). A vertical datum defines elevations relative to a reference surface/geoid (e.g., NAVD88).

What is the difference between NAD27 and NAD83 datums?

NAD27 uses the Clarke 1866 ellipsoid with a non-geocentric origin (Meades Ranch, Kansas). NAD83 uses the GRS80 ellipsoid and is essentially Earth-centered (geocentric), giving different coordinates for the same physical point.

Define geodetic latitude and longitude on the reference ellipsoid.

Geodetic latitude is the angle between the ellipsoid normal at a point and the equatorial plane; geodetic longitude is the angle, measured in the equatorial plane, from the prime meridian to the point's meridian.

What is the State Plane Coordinate System (SPCS), and which two map projections does it use?

SPCS is a U.S. system of plane (X/Y) coordinates dividing states into zones to limit distortion. It uses the Transverse Mercator projection for north–south elongated zones and the Lambert Conformal Conic projection for east–west elongated zones (Oblique Mercator for the Alaska panhandle).

In the UTM coordinate system, how wide is each zone and what are easting/northing conventions?

UTM uses $6°$-wide longitudinal zones (60 total). The central meridian is assigned a false easting of $500{,}000$ m; northing is measured from the equator ($0$ m in the Northern Hemisphere, $10{,}000{,}000$ m false northing in the Southern Hemisphere).

Define the convergence of meridians and its significance for azimuths.

Convergence is the angle by which grid north differs from true (geodetic) north because meridians converge toward the poles. It causes a difference between geodetic azimuth and grid azimuth, related by $\text{grid azimuth} = \text{geodetic azimuth} - \gamma$ (the convergence angle $\gamma$).

Write the combined formula for the correction due to Earth curvature and atmospheric refraction in leveling, in U.S. units (feet, with distance $K$ in miles).

$$h_{cr} = 0.0206\,K^{2}\ \text{ft}$$ where $K$ is the sight distance in miles (curvature $0.0239K^2$ minus refraction $\approx 0.0033K^2$).

How do the effects of Earth curvature and atmospheric refraction differ in sign in differential leveling?

Curvature makes a level (horizontal) sight read too high (objects appear lower), increasing the rod reading; refraction bends the line of sight downward, partially offsetting curvature by about one-seventh. The net effect still lowers the apparent elevation of distant points.

In trigonometric leveling/long sights, what is the approximate combined curvature-and-refraction correction in metric units with distance $K$ in kilometers?

$$h_{cr} \approx 0.0675\,K^{2}\ \text{m}$$ where $K$ is in kilometers (curvature $0.0785K^2$ minus refraction $\approx 0.0110K^2$).

Why is balancing backsight and foresight distances important in differential leveling?

Equal backsight and foresight distances cancel the systematic effects of Earth curvature, atmospheric refraction, and instrument collimation error, since these effects are equal and opposite on the two readings.