GELE Adjustment Computations (Least Squares) — Theory of Errors, Weights and Most Probable ValueRevision Notes
Revision notes for GELE Adjustment Computations (Least Squares) — Theory of Errors, Weights and Most Probable Value. Short, focused, and designed for the week before exam day. Use these when you are already familiar with the chapter and need a quick refresh on the high-yield items Professional Regulation Commission (PRC) — Board of Geodetic Engineering tests.
Exam context
On the GELE 2026, the Adjustment Computations (Least Squares) subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Geodetic Engineering's pattern. Theory of Errors, Weights and Most Probable Value lands at position 1st out of 5 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Adjustment Computations (Least Squares) on a typical GELE paper.
Theory of Errors, Weights and Most Probable Value - Revision Notes
Adjustment Computations begins with a fundamental truth: every measurement contains error. When a geodetic engineer measures a distance, angle, or elevation difference, the observed value is never perfectly exact. The theory of errors provides the mathematical framework for understanding, classifying, and handling these inevitable imperfections. This chapter establishes the conceptual and computational foundation for least squares adjustment — the gold standard method used in all professional surveying and geodetic work in the Philippines and worldwide. Mastery of error types, the most probable value (MPV), weights, and the weighted mean is essential not only for the PRC Geodetic Engineer Licensure Examination but for every professional task you will encounter in the field — from NAMRIA geodetic control surveys to cadastral surveys under PD 1529 and land titling work governed by CA 141.
Sections
Exam Tips
- Board exams often ask you to classify a given source of error — memorize the three types with one concrete surveying example each.
- The phrase 'errors that cancel out with repetition' always refers to random errors, not systematic errors.
- If a question mentions 'normal distribution,' 'Gaussian,' or 'probability,' it is referring to random errors.
- Systematic errors are sometimes called 'cumulative errors' in older textbooks — recognize both terms.
Key Points
- All measurements contain errors — perfection is physically impossible due to instrument limitations, human factors, and natural conditions.
- Errors are classified into three types: blunders (mistakes), systematic errors, and random (accidental) errors.
- Blunders are large, careless errors that must be detected and eliminated before any adjustment is performed.
- Systematic errors follow a definable mathematical pattern and can be modeled and corrected using calibration or field procedures.
- Random errors are small, unpredictable, and vary in sign — they remain after blunders are removed and systematic errors are corrected.
- Random errors obey the normal (Gaussian) probability distribution — this statistical behavior justifies the use of least squares adjustment.
- The four properties of random errors (Gauss's postulates): small errors are more frequent than large; positive and negative errors of the same magnitude are equally likely; very large errors rarely occur; the most probable value minimizes the sum of squared residuals.
- Adjustment computations deal exclusively with random errors — blunders and systematic errors must be handled before adjustment.
Definitions
Term
Blunder (Mistake)
Definition
A gross error caused by carelessness, misreading of instruments, incorrect booking, or miscommunication. Examples include transposing digits (reading 53.24 m as 52.34 m) or misidentifying a target point.
Importance
Blunders must be identified and eliminated before adjustment. Including a blundered observation in a least squares solution corrupts all adjusted results.
Term
Systematic Error
Definition
An error that follows a consistent pattern related to a known cause — it accumulates with the quantity measured. Examples: EDM constant offset, refraction correction, sea-level correction for distances on PRS92 control networks.
Importance
Systematic errors are modeled and removed through calibration, correction formulas, or field procedures (e.g., reciprocal vertical angles, double-centering). They do not average out with more observations.
Term
Random (Accidental) Error
Definition
The small, residual error remaining after blunders are removed and systematic errors are corrected. Random errors are governed by the Gaussian normal distribution.
Importance
Random errors are the subject of adjustment computations. Their statistical properties justify the method of least squares — the mathematical engine of all professional geodetic adjustment.
Term
Residual (v)
Definition
The difference between an individual observation and the most probable value: v_i = x_i − x̄. Residuals are the computed quantities in adjustment; true errors are never known.
Importance
Residuals are used to compute standard deviations and to assess the quality of an adjustment. Large residuals flag potential blunders.
Section Title
Classification of Errors
Common Mistakes
- Confusing residuals with true errors — true errors are never known; only residuals can be computed from observations.
- Attempting to adjust data that still contains blunders — always perform outlier detection (e.g., comparing against closure tolerances in PD 1529 surveys) before adjustment.
- Believing that more observations eliminate systematic errors — systematic errors accumulate regardless of the number of repetitions.
- Forgetting that random errors can be positive or negative — the mean of random errors approaches zero as the number of observations increases.
Formulas
Example
Five distance measurements: 45.231, 45.228, 45.235, 45.229, 45.232 m. x̄ = (45.231+45.228+45.235+45.229+45.232)/5 = 226.155/5 = 45.231 m.
Formula
x̄ = Σxᵢ / n
Variables
x̄ = arithmetic mean (MPV); xᵢ = individual observation; n = total number of observations
Application
Computes the most probable value for a set of equally reliable direct observations. Used when all observations have the same precision (equal weights).
Example
For x̄ = 45.231 m and x₁ = 45.228 m: v₁ = 45.228 − 45.231 = −0.003 m.
Formula
vᵢ = xᵢ − x̄
Variables
vᵢ = residual of observation i; xᵢ = individual observation; x̄ = arithmetic mean
Application
Computes the residual for each observation. Note: Σvᵢ = 0 always (arithmetic check). The sum of residuals equals zero for a correct mean calculation.
Example
For the five distances above, Σv² = (0.000)²+(−0.003)²+(0.004)²+(−0.002)²+(0.001)² = 0+9+16+4+1 = 30 (×10⁻⁶ m²). σ = √(30×10⁻⁶/4) = √(7.5×10⁻⁶) = 0.00274 m ≈ 0.003 m.
Formula
σ = √(Σv² / (n−1))
Variables
σ = standard deviation of a single observation; v = residual (xᵢ − x̄); n = number of observations; (n−1) = degrees of freedom
Application
Measures the dispersion of individual observations around the mean. Uses n−1 (not n) because one degree of freedom is consumed in estimating the mean from the sample.
Example
With σ = 0.00274 m and n = 5: σ_x̄ = 0.00274/√5 = 0.00274/2.236 = 0.00122 m ≈ 0.001 m.
Formula
σ_x̄ = σ / √n
Variables
σ_x̄ = standard deviation of the mean; σ = standard deviation of a single observation; n = number of observations
Application
Measures the precision of the computed mean. Averaging multiple observations improves precision. This formula is the mathematical basis for taking multiple readings in precision surveying.
Example
With σ = 0.00274 m: PE = 0.6745 × 0.00274 = 0.00185 m ≈ 0.002 m.
Formula
PE = 0.6745 σ
Variables
PE = probable error of a single observation; σ = standard deviation of a single observation
Application
The probable error defines the range (±PE) within which there is a 50% probability that the true error lies. Useful in older surveying literature and some board exam problems.
Exam Tips
- The check Σv = 0 is a free arithmetic verification — always perform it after computing residuals.
- Board problems often give σ and ask for σ_x̄ or vice versa — the link is always √n.
- PE = 0.6745σ appears in problems where 50% confidence intervals are requested — memorize the coefficient 0.6745.
- If a problem states 'observations of equal reliability' or 'equally precise,' use arithmetic mean and σ = √(Σv²/(n−1)).
- The phrase 'best estimate' in a board problem always means the MPV (arithmetic mean for equal weights).
Key Points
- For a set of equally reliable (equal-weight) direct observations, the Most Probable Value (MPV) is the arithmetic mean.
- The arithmetic mean minimizes the sum of squared residuals — this is the least squares criterion for direct observations.
- The standard deviation of a single observation (σ) describes the precision of any one measurement in the set.
- The standard deviation of the mean (σ_x̄) describes the precision of the computed mean — it is smaller than σ by a factor of √n.
- The probable error (PE) is the error that has a 50% probability of being exceeded — it equals 0.6745σ.
- Increasing the number of observations improves the precision of the mean by √n, not by n — doubling precision requires quadrupling observations.
- The degree of freedom (redundancy) for direct observations equals n − 1, where n is the number of observations.
Definitions
Term
Most Probable Value (MPV)
Definition
The value of a measured quantity that is statistically most likely to be closest to the true value, given the set of observations. For equally reliable observations, it is the arithmetic mean.
Importance
The MPV is the central output of any adjustment computation. In least squares, the MPV minimizes the weighted sum of squared residuals — ΣwᵢVᵢ² = minimum.
Term
Standard Deviation (σ)
Definition
The square root of the average squared deviation from the mean (using n−1 for samples). It measures the spread or precision of individual observations.
Importance
σ is the primary precision indicator in geodetic work. NAMRIA, DENR, and international standards specify acceptable σ values for different survey orders (1st, 2nd, 3rd order).
Term
Probable Error (PE)
Definition
The magnitude of error that is equally likely to be exceeded or not exceeded — 50th percentile of the absolute error distribution. PE = 0.6745σ.
Importance
Appears frequently in older board exam problems and in some Philippine survey specifications. Know its relationship to σ.
Term
Degrees of Freedom
Definition
The number of independent pieces of information available for estimating statistical parameters. For n direct observations estimating one mean, degrees of freedom = n − 1.
Importance
Using n instead of n−1 in the denominator of σ underestimates variability — a common board exam trap. Always use n−1 for sample standard deviation.
Section Title
Most Probable Value (MPV) and Standard Deviation
Common Mistakes
- Using n instead of n−1 in the denominator of σ — the correct formula is σ = √(Σv²/(n−1)) for sample data.
- Confusing σ (precision of one observation) with σ_x̄ (precision of the mean) — σ_x̄ is always smaller than σ.
- Forgetting to verify Σvᵢ = 0 as an arithmetic check after computing residuals.
- Misidentifying the MPV as simply the 'middle value' (median) — for symmetric distributions, mean and median coincide, but the MPV is always the mean (minimizes Σv²).
- Believing that quadrupling observations doubles the standard deviation — it actually halves σ_x̄ (improvement is by √n).
Formulas
Example
Obs. 1: σ₁ = 0.02 m → w₁ = 1/0.02² = 1/0.0004 = 2500. Obs. 2: σ₂ = 0.01 m → w₂ = 1/0.01² = 1/0.0001 = 10000. Weight ratio: w₁:w₂ = 2500:10000 = 1:4.
Formula
wᵢ = 1 / σᵢ²
Variables
wᵢ = weight of observation i; σᵢ = standard deviation of observation i
Application
Fundamental weight assignment from precision. A measurement with half the standard deviation receives four times the weight — it is four times more precise in terms of variance.
Example
From above: x̄_w = (2500×100.02 + 10000×100.05)/(2500+10000) = (250050+1000500)/12500 = 1250550/12500 = 100.044 m.
Formula
x̄_w = Σ(wᵢ xᵢ) / Σwᵢ
Variables
x̄_w = weighted mean; wᵢ = weight of observation i; xᵢ = individual observation value
Application
Computes the MPV when observations have unequal weights. This is the generalization of the arithmetic mean — when all weights are equal, it reduces to the simple mean.
Example
Three routes to a benchmark: K₁ = 2 km, K₂ = 1 km, K₃ = 4 km. Relative weights: w₁ = 1/2 = 0.5, w₂ = 1/1 = 1.0, w₃ = 1/4 = 0.25. Or integer form (multiply by 4): w₁:w₂:w₃ = 2:4:1.
Formula
w ∝ 1/K (Leveling)
Variables
w = weight of a leveling line; K = length of the leveling route (in km)
Application
In differential leveling, errors accumulate proportionally to distance. A 2 km route accumulates more error than a 1 km route, so shorter routes receive higher weights.
Example
Value A is the mean of 4 readings, Value B is the mean of 9 readings (both same-precision instrument). w_A:w_B = 4:9.
Formula
w ∝ n (Repeated Observations)
Variables
w = weight; n = number of observations averaged to produce the reported value
Application
When a reported value is the mean of n individual measurements, its precision improves by √n (σ_x̄ = σ/√n), so its weight (∝ 1/σ²) increases by factor n.
Exam Tips
- MEMORIZE: w = 1/σ² (NOT 1/σ). This distinction is tested every exam cycle.
- For leveling weight problems: list the km distances, take reciprocals, scale to integers for easy computation.
- The weighted mean always lies between the minimum and maximum observed values — use this as a quick sanity check.
- If one weight is much larger than others, the weighted mean will be pulled strongly toward that observation — verify this intuitively.
- When weights are given as ratios (e.g., 1:2:3), you can use them directly in Σwx/Σw without converting to σ values.
- For problems mixing GPS (high precision) with total station (lower precision), always assign GPS observations higher weights.
Key Points
- Weight is a dimensionless number expressing the relative reliability of an observation compared to others in the same group.
- Weight is inversely proportional to variance: wᵢ = c/σᵢ² where c is any positive constant (often set to 1).
- When observations have different precision levels, the weighted mean gives the MPV — it assigns more influence to more precise (lower σ) observations.
- For differential leveling, weight is inversely proportional to the length (km) of the leveling route: w ∝ 1/K.
- For repeated measurements of equal precision, weight is directly proportional to the number of repetitions: w ∝ n.
- Relative weights — not absolute weights — determine the weighted mean. You may multiply or divide all weights by the same constant without changing the result.
- The standard deviation of the weighted mean uses the weighted residuals in its computation.
Definitions
Term
Weight (w)
Definition
A positive dimensionless number assigned to each observation expressing its reliability relative to other observations in the same adjustment. Higher weight = more reliable = lower variance.
Importance
Weights are the mechanism by which least squares incorporates heterogeneous data (e.g., mixing GPS vectors of different baseline lengths with terrestrial measurements in a Philippine PPCS adjustment).
Term
Weighted Mean (x̄_w)
Definition
The most probable value for a set of observations of unequal reliability. It is the weighted average: Σ(wᵢxᵢ)/Σwᵢ.
Importance
The weighted mean is the foundation of weighted least squares. When the normal equations of a least squares system are solved, the result is equivalent to computing weighted means for each unknown.
Term
Unit Weight
Definition
A reference weight of 1 (unity) assigned to a baseline observation. Other weights are expressed relative to this reference. The corresponding variance is called the reference variance or variance of unit weight (σ₀²).
Importance
The variance of unit weight (σ₀²) is the key quality indicator in full least squares adjustments. It is reported in NAMRIA geodetic control network adjustments.
Section Title
Weights and Weighted Mean
Common Mistakes
- Computing w = 1/σ instead of w = 1/σ² — this is the single most common board exam error in weight problems. Weight is inversely proportional to VARIANCE, not standard deviation.
- For leveling problems, using w ∝ K (direct proportion) instead of w ∝ 1/K — longer routes have LOWER weights.
- Mixing up which observation gets the higher weight: higher weight → lower variance → higher precision → smaller σ.
- Using absolute weights when only relative weights are needed — any consistent set of relative weights gives the same weighted mean.
- Forgetting to verify the weighted mean by checking it lies between the smallest and largest observed values.
Formulas
Example
Values in seconds above 35°24'00": 10, 15, 8, 12, 14, 11. Mean = 70/6 = 11.667" ≈ 11.7". MPV = 35°24'11.7". Residuals: −1.7, +3.3, −3.7, +0.3, +2.3, −0.7. Σv² = 2.89+10.89+13.69+0.09+5.29+0.49 = 33.34 sq-sec. σ = √(33.34/5) = √6.668 = 2.58" ≈ 2.6".
Formula
Board Problem 1 — MPV and σ
Variables
Six angles measured: 35°24'10", 35°24'15", 35°24'08", 35°24'12", 35°24'14", 35°24'11". Find MPV and σ.
Application
Step 1: Convert to seconds for easy arithmetic. Step 2: Compute mean. Step 3: Compute residuals. Step 4: Compute Σv². Step 5: σ = √(Σv²/(n−1)).
Example
w₁=1/0.0009=1111.1, w₂=1/0.0004=2500.0, w₃=1/0.0004=2500.0. Σw=6111.1. Σwx=1111.1×150.32+2500×150.28+2500×150.30=166,922.5+375,700+375,750=918,372.5. x̄_w=918372.5/6111.1=150.284 m ≈ 150.28 m. Note: The two more precise observations (w=2500 each) dominate the result.
Formula
Board Problem 2 — Weighted Mean
Variables
A distance measured three times: 150.32 m (σ=0.03 m), 150.28 m (σ=0.02 m), 150.30 m (σ=0.02 m). Find the weighted mean.
Application
Step 1: Compute weights w = 1/σ². Step 2: Compute wx products. Step 3: Weighted mean = Σwx/Σw.
Example
w_A=1/3, w_B=1/5, w_C=1/2. Multiply by 30 (LCM): w_A=10, w_B=6, w_C=15. Σw=31. Σwx=10×14.235+6×14.228+15×14.241=142.35+85.368+213.615=441.333. x̄_w=441.333/31=14.236 m. Route C (shortest, highest weight) pulls the result toward 14.241 m.
Formula
Board Problem 3 — Leveling Weights
Variables
Three leveling routes to BM-Pasig (a fictitious NAMRIA-style benchmark): Route A = 3 km, gives elev. 14.235 m; Route B = 5 km, gives 14.228 m; Route C = 2 km, gives 14.241 m. Find the best estimate of the elevation.
Application
Step 1: Assign w ∝ 1/K. Step 2: Use integer weights (multiply by LCM of denominators). Step 3: Compute weighted mean elevation.
Exam Tips
- Set up a table for all weight/weighted-mean problems: Column 1 = observation, Column 2 = σ, Column 3 = σ², Column 4 = w = 1/σ², Column 5 = wx. Sum columns 4 and 5, then divide.
- For angle observations, convert everything to seconds, compute the mean in seconds, then convert back to DMS for the final answer.
- In leveling problems, the LCM trick for integer weights saves time and reduces rounding errors.
- After computing the weighted mean, always ask: 'Does this value make sense? Is it between the min and max observations?' If not, recheck your weights.
Key Points
- Board problems on this topic fall into three categories: (1) MPV and σ from direct observations, (2) weighted mean from unequal-precision observations, and (3) weight assignment for leveling routes.
- Always show the residual table (observation, mean, residual, residual²) for MPV/σ problems.
- For weighted mean problems, show the weight computation table (σ, σ², w, wx) explicitly.
- Verify Σv = 0 for equal-weight problems; there is no equivalent simple check for weighted residuals.
- Read problems carefully — 'standard deviation of the mean' is σ_x̄ = σ/√n, not σ.
Section Title
Board-Exam Worked Problems
Common Mistakes
- In the angle problem, working in degrees-minutes-seconds without converting to a single unit — always convert to total seconds before computing the mean.
- Rounding intermediate weights to integers prematurely — keep full decimal precision in weights until the final weighted mean is computed.
- In leveling problems, forgetting to check that the weighted mean lies between the minimum (14.228 m) and maximum (14.241 m) observed elevations.
- Misreading 'standard deviation of the mean' as σ — it is σ/√n, which is always smaller.
Formulas
Example
Given σ = 0.015 m for a single tape measurement: σ_x̄ for 9 repetitions = 0.015/3 = 0.005 m. PE = 0.6745×0.015 = 0.0101 m. Weight relative to a measurement with σ = 0.030 m: w ratio = (0.030)²/(0.015)² = 0.0009/0.000225 = 4. The more precise measurement has 4× the weight.
Formula
Summary: x̄ = Σx/n | σ = √(Σv²/(n−1)) | σ_x̄ = σ/√n | PE = 0.6745σ | wᵢ = 1/σᵢ² | x̄_w = Σwx/Σw
Variables
All symbols as defined in previous sections
Application
Complete formula set for this chapter. These six formulas cover all board exam question types under Theory of Errors and Weights.
Exam Tips
- Write the six core formulas on your exam scratch paper at the start — this prevents memory lapses under pressure.
- The 0.6745 coefficient: memorize it as 'roughly 2/3' for quick estimation, but use the exact value (0.6745) in calculations.
- When a problem asks for '90% confidence interval,' use 1.645σ; for '95%,' use 1.96σ — these may appear in advanced board items.
- A quick way to remember PE vs σ: PE < σ always (since 0.6745 < 1). If your answer gives PE > σ, you've made an error.
Key Points
- Know all five core formulas: arithmetic mean, residual, standard deviation of observation, standard deviation of mean, and weighted mean.
- Understand the hierarchy: x̄_w (MPV) → vᵢ (residuals) → σ (precision of one obs.) → σ_x̄ (precision of mean) → PE (probable error).
- The factor 0.6745 converts σ to PE; the factor √n converts σ to σ_x̄.
- The relationship between weight and standard deviation: wᵢ = 1/σᵢ² means doubling σ reduces weight by factor 4.
- The 68-95-99.7 rule for the normal distribution: 68% of observations fall within ±1σ, 95% within ±2σ, 99.7% within ±3σ.
Definitions
Term
Variance (σ²)
Definition
The square of the standard deviation. The average of squared deviations from the mean (using n−1). Weight is inversely proportional to variance.
Importance
Variance is additive for independent errors — the variance of a sum equals the sum of individual variances. This property is the basis of error propagation in least squares.
Term
Normal Distribution
Definition
The bell-shaped probability distribution that describes random errors. Characterized completely by its mean (μ) and standard deviation (σ). The distribution is symmetric about the mean.
Importance
The assumption of normally distributed errors is what makes the arithmetic mean the MPV and justifies the least squares criterion (minimizing Σv² maximizes the likelihood of the observed data).
Section Title
Key Formulas Summary and Error Measures Comparison
Common Mistakes
- Forgetting the 0.6745 factor for probable error — PE ≠ σ.
- Confusing the 68-95-99.7 rule percentages: 68% (±1σ), 95% (±2σ), 99.7% (±3σ).
- Applying σ_x̄ = σ/√n but forgetting that n must be the number of INDEPENDENT observations of equal precision.
Connections
- Theory of Errors → Least Squares Adjustment: The properties of random errors (normal distribution, zero mean) mathematically justify the least squares criterion (minimize Σwv²). This chapter is the prerequisite for all subsequent LS topics.
- Weights (w = 1/σ²) → Normal Equations: In matrix least squares (A^T W A x = A^T W L), the weight matrix W = diag(w₁, w₂, ..., wₙ) is built directly from the individual weights defined in this chapter.
- Leveling Weights (w ∝ 1/K) → Differential Leveling Networks: When adjusting a Philippine geodetic leveling network (e.g., NAMRIA 1st-order leveling), each leveling section is assigned weight 1/K. The resulting adjustment yields the MPV elevation for each benchmark.
- Standard Deviation of Mean → Survey Specifications: PD 1529 and DENR survey regulations specify precision requirements in terms of standard deviation. Understanding σ_x̄ = σ/√n helps determine how many repetitions are needed to meet a required precision.
- Weighted Mean → Traverse Adjustment: In compass rule and transit rule traverse adjustment, the underlying concept of distributing misclosures proportionally is a simplified form of weighted adjustment.
- Error Theory → GPS/GNSS Processing: In Philippine PPCS/UTM coordinate computations using GPS, different baselines have different standard deviations based on baseline length and observation time. The weight matrix for GPS adjustment is W = C⁻¹ (inverse of the covariance matrix), directly applying w = 1/σ².
- Probable Error → Survey Grade Standards: Some older NAMRIA and BLM (Bureau of Lands and Management, now LMB) documents express accuracy requirements in terms of probable error. Understanding PE = 0.6745σ bridges old and new standards.
- Normal Distribution → Outlier Detection: The 3σ rule (±3σ encompasses 99.7% of observations) is used as a blunder detection criterion — any residual exceeding 3σ is flagged as a potential blunder before the final adjustment.
- MPV Concept → Traverse Closure: The angular and linear misclosures in a traverse represent the 'residual' of the network. Distributing these misclosures to find the adjusted coordinates is an application of the MPV concept at the network level.
- RA 8560 (Geodetic Engineering Act) → Professional Responsibility: As licensed geodetic engineers, you are legally responsible for the accuracy of survey results. Understanding error theory is not just academic — it is the technical basis for meeting the professional and legal standards required under RA 8560 and the surveys governed by PD 1529 and CA 141.
Exam Strategy
For the Theory of Errors and Weights section of the PRC Geodetic Engineer Licensure Examination, expect 5–10 questions covering: (1) error classification (1–2 items — conceptual, choose the type of error), (2) MPV and standard deviation from direct observations (2–3 items — numerical, set up residual table), (3) weighted mean from unequal-precision observations (2–3 items — numerical, set up weight table), and (4) leveling route weights (1–2 items). Allocate approximately 3–5 minutes per numerical problem. Start by writing the six core formulas on your scratch paper. For any problem involving weights, immediately set up a four-column table: (obs. value | σ | σ² | w = 1/σ²). For leveling problems, list distances, take reciprocals, and multiply to get integer weights. Always verify: (a) Σv = 0 for equal-weight problems, and (b) the weighted mean falls between the min and max observations. The most tested trap is w = 1/σ vs. w = 1/σ² — always write 'w = 1/σ²' first before doing any calculation. For multi-step problems, never round intermediate values (keep at least 4 significant figures in weights and products). The probable error formula PE = 0.6745σ appears less frequently but is easily answered if the factor 0.6745 is memorized. Know the conceptual reasoning behind each formula — board examiners often test the 'why' (e.g., 'why is w ∝ 1/K for leveling?') not just the 'how'. Review the three worked examples in the reference notes until you can solve them from memory in under 3 minutes each.
Quick Review Questions
A distance is measured six times with the following values: 125.32, 125.28, 125.35, 125.30, 125.29, 125.31 m. What is the Most Probable Value (MPV)?
MPV = arithmetic mean = (125.32+125.28+125.35+125.30+125.29+125.31)/6 = 751.85/6 = 125.308 m. For equally reliable observations, the MPV is always the arithmetic mean.
Using the six measurements above (MPV = 125.308 m), compute the standard deviation of a single observation.
Residuals: +0.012, −0.028, +0.042, −0.008, −0.018, +0.002. Σv² = 0.000144+0.000784+0.001764+0.000064+0.000324+0.000004 = 0.003084 m². σ = √(0.003084/5) = √0.0006168 = 0.02484 m ≈ 0.023 m (or 0.025 m depending on rounding of MPV).
What is the standard deviation of the mean for the six measurements if σ = 0.025 m?
σ_x̄ = σ/√n = 0.025/√6 = 0.025/2.449 = 0.0102 m. The mean of six observations is about 2.45 times more precise than a single observation.
An observation has σ = 0.030 m. A second observation of the same quantity has σ = 0.015 m. What is the weight ratio w₁:w₂?
w = 1/σ². w₁ = 1/0.030² = 1/0.0009 = 1111. w₂ = 1/0.015² = 1/0.000225 = 4444. Ratio = 1111:4444 = 1:4. Halving the standard deviation quadruples the weight because weight is proportional to 1/σ².
Three leveling routes to a benchmark have lengths of 4 km, 2 km, and 8 km with respective elevations of 52.318, 52.325, and 52.312 m. What are the relative (integer) weights?
w ∝ 1/K: w₁ = 1/4 = 0.25, w₂ = 1/2 = 0.50, w₃ = 1/8 = 0.125. Multiply by 8 (LCM): w₁ = 2, w₂ = 4, w₃ = 1. The 2 km route (shortest) has the highest weight.
Using the leveling data above (w₁=2, w₂=4, w₃=1, elevations 52.318, 52.325, 52.312 m), find the weighted mean elevation.
Σw = 2+4+1 = 7. Σwx = 2×52.318+4×52.325+1×52.312 = 104.636+209.300+52.312 = 366.248. x̄_w = 366.248/7 = 52.321 m. The 2 km route (w=4) pulls the result toward 52.325 m.
What is the probable error of a single observation if σ = 0.018 m?
PE = 0.6745 × σ = 0.6745 × 0.018 = 0.01214 m ≈ 0.012 m. This means there is a 50% probability that the error of any single observation is less than ±0.012 m.
How many observations are needed to reduce σ_x̄ to 1/3 of σ (the single-observation standard deviation)?
σ_x̄ = σ/√n. For σ_x̄ = σ/3: σ/√n = σ/3 → √n = 3 → n = 9. This illustrates the diminishing returns of adding more observations — to triple the precision of the mean, you need 9 times as many observations.
In an adjustment computation, why must blunders be removed BEFORE applying least squares?
Least squares minimizes Σwv². If a blunder is present (e.g., an observation that is off by 1 m when all others agree to ±1 mm), the squared residual for that observation (1000² = 1,000,000 mm²) overwhelms all other residuals (each ≈1 mm²). The adjustment will shift all adjusted values toward the blundered observation, introducing errors into everything.
A value reported as the mean of 4 equal-precision readings is to be compared with a value from a single reading of the same instrument. What is the weight ratio (group mean : single reading)?
For the mean of 4 readings: σ_x̄ = σ/√4 = σ/2. Weight = 1/(σ/2)² = 4/σ² = 4 × (1/σ²). For the single reading: weight = 1/σ². Ratio = 4:1. This confirms w ∝ n: weight is directly proportional to the number of observations averaged.
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