Theory of Errors, Weights and Most Probable Value - Cheat Sheet
Your last-minute rapid-fire reference for error theory, weights, and most probable values in adjustment computations. Focus on formulas, definitions, and exam traps—zero fluff.
Error Classification & Characteristics
Common values
Value
0.6745
Symbol
—
Quantity
Probable Error constant
Important facts
- Blunders must be detected and rejected—no statistical cure.
- Systematic errors must be modeled, corrected, or eliminated before least squares.
- Random errors follow Gaussian (normal) distribution; adjustment assumes they are random and independent.
- Standard deviation σ and variance σ² quantify random error magnitude.
- Probable error PE = 0.6745σ (50% of errors fall within ±PE).
Key definitions
Blunder (Mistake)
Example
Misreading a rod as 1.53 m instead of 2.53 m
Large, identifiable error from human carelessness or instrument malfunction—always eliminate before adjustment.
Systematic Error
Example
Tape always reads 0.05 m too short due to temperature effect
Consistent, directional error due to instrument or environment—model and correct mathematically before adjustment.
Random Error (Accidental Error)
Example
Slight centering misalignment varying ±0.001 m per setup
Small, unpredictable, sign-varying errors governed by normal distribution—handled by adjustment and statistical methods.
Diagrams to know
- Bell curve (normal distribution) with σ, 2σ, 3σ marked
- Error classification flowchart: Identify → Blunder? → Eliminate; Systematic? → Correct; Random? → Adjust
Most Probable Value (MPV) for Equal-Weight Observations
Formulas
x̄ = Σx / n
Meaning
x̄ = arithmetic mean; Σx = sum of all observations; n = number of observations
Watch out
Do NOT use this if observations have different precisions—use weighted mean instead.
When to use
When all observations have equal reliability (equal weight).
vᵢ = xᵢ − x̄
Meaning
vᵢ = residual of observation i; xᵢ = individual observation; x̄ = arithmetic mean
Watch out
Residuals are deviations from the MPV, not errors. Sum of residuals must equal zero (within rounding).
When to use
Always compute residuals to verify Σv ≈ 0 and calculate standard deviation.
σ = √(Σv² / (n−1))
Meaning
σ = standard deviation of a single observation; Σv² = sum of squared residuals; n−1 = degrees of freedom
Watch out
Use n−1, NOT n. Using n underestimates error. This is sample, not population, standard deviation.
When to use
Quantify the precision of individual measurements; use n−1 (sample standard deviation).
σₓ̄ = σ / √n
Meaning
σₓ̄ = standard deviation of the mean; σ = single-observation std dev; n = number of observations
Watch out
Improvement is √n, NOT n. Adding 4 measurements reduces error by factor of 2, not 4.
When to use
Estimate precision of the final MPV result; shows how many measurements improve accuracy.
Important facts
- For equal-weight observations, the arithmetic mean is the most probable value.
- Σv = 0 always (within rounding error)—use as a check on calculations.
- Σv² quantifies total error; larger Σv² means poorer precision.
- Standard deviation σ represents precision of individual observations.
- Standard deviation of mean σₓ̄ is always smaller than σ by factor √n.
- n−1 is used because one degree of freedom is lost estimating the mean.
- Larger n always improves precision (reduces σₓ̄), but only by √n, not linearly.
Key definitions
Arithmetic Mean
Example
Five distance readings: 100.01, 100.02, 100.03, 100.02, 100.02 m → mean = 100.02 m
Sum of all equal-weight observations divided by count—the most probable value for redundant equal-precision data.
Residual
Example
If x̄ = 100.02 m and one reading is 100.03 m, residual v = +0.01 m
Difference between an observation and the most probable value; v = x − x̄.
Standard Deviation (σ)
Example
σ = 0.015 m means most readings fall within ±0.015 m of the mean
Root-mean-square of residuals; measures scatter and precision of observations around the mean.
Diagrams to know
- Residual scatter plot around mean showing positive and negative deviations
- Precision improvement curve: σₓ̄ vs. n (hyperbolic decay)
Weights & Weight Assignment Rules
Formulas
wᵢ = 1 / σᵢ²
Meaning
wᵢ = weight of observation i; σᵢ² = variance (std dev squared) of observation i
Watch out
Weight is INVERSE to σ², not σ. Never use 1/σ or 1/σᵢ—always 1/σ².
When to use
Assign weights when observations have different precisions; inverse proportional to variance.
wᵢ ∝ 1/Kᵢ (leveling)
Meaning
wᵢ = weight; Kᵢ = distance (in km) of leveling route i
Watch out
Weight is INVERSE to distance. A 1 km route is 2× more reliable than a 2 km route; assign w = 2 vs. w = 1.
When to use
In leveling networks, shorter routes are more reliable and get higher weight.
wᵢ ∝ nᵢ (repeated measurements)
Meaning
wᵢ = weight; nᵢ = number of repeated measurements for observation i
Watch out
Higher n = higher weight (opposite of distance rule). More repeats = more reliable.
When to use
When an observation is repeated multiple times, weight is directly proportional to repetition count.
Important facts
- Weight w = 1/σ²—always reciprocal of VARIANCE, not standard deviation.
- Higher precision (smaller σ) → higher weight (more influence on MPV).
- In leveling: shorter distance → shorter random error → higher weight; w ∝ 1/K.
- More repetitions → higher weight; w ∝ n (directly proportional).
- Weight can be normalized (relative weight) without changing final result.
- Zero weight means zero influence on weighted mean; should not occur in practice.
Key definitions
Weight
Example
Observation A with σ = 0.02 m has weight wₐ = 1/0.02² = 2500; observation B with σ = 0.01 m has wᵦ = 10,000 (B is more reliable, higher weight)
A dimensionless number expressing relative reliability of an observation; inversely proportional to variance (error²).
Relative Weight
Example
If computed weights are 2500 and 10,000, relative weights are 1 and 4 (divide by 2500)
Weight normalized to a reference; often used to compare observations without absolute variance values.
Diagrams to know
- Weight hierarchy chart showing precision vs. weight relationship
- Leveling network with weights assigned by route distance
Weighted Mean & Unequal-Weight Observations
Formulas
x̄w = Σ(wᵢ·xᵢ) / Σwᵢ
Meaning
x̄w = weighted mean; wᵢ = weight of observation i; xᵢ = observation value; Σwᵢ = sum of all weights
Watch out
Numerator is SUM OF WEIGHTED OBSERVATIONS, not sum of weights times mean. Every xᵢ must be multiplied by its own wᵢ.
When to use
When observations have different weights (unequal precision); always use for least squares foundation.
vᵢ = xᵢ − x̄w
Meaning
vᵢ = weighted residual; xᵢ = observation; x̄w = weighted mean
Watch out
Residuals from weighted mean do NOT sum to zero in general (unlike unweighted case). Use Σ(wᵢ·vᵢ) ≈ 0 as check.
When to use
Compute residuals around weighted mean for error analysis and standard deviation of weighted mean.
σw = √(Σ(wᵢ·vᵢ²) / (n−1))
Meaning
σw = standard deviation of weighted observations; wᵢ·vᵢ² = weighted squared residual; n−1 = degrees of freedom
Watch out
Use n−1 for sample; numerator must include weights (wᵢ·vᵢ², NOT just vᵢ²).
When to use
Measure precision of individual weighted observations.
σx̄w = σw / √(Σwᵢ)
Meaning
σx̄w = standard deviation of weighted mean; σw = weighted single-observation std dev; Σwᵢ = sum of all weights
Watch out
Denominator is NOT √n; it is √(Σwᵢ). With unequal weights, use sum of weights, not count.
When to use
Estimate precision of the final weighted-mean result.
Important facts
- Higher weight observations have greater influence on weighted mean.
- More reliable (lower σ) measurements pull the result toward their value.
- Weighted mean always lies between minimum and maximum observation values.
- Σ(wᵢ·vᵢ) ≈ 0 should hold for unweighted data as internal consistency check.
- Standard deviation of weighted mean depends on Σwᵢ, not n.
- Relative weights (scale factor) do NOT change the weighted mean value.
- Weight assignment errors directly affect final result; double-check w = 1/σ² calculation.
Key definitions
Weighted Mean
Example
Two distance measurements: 100.02 m (σ=0.02) and 100.05 m (σ=0.01) → weighted mean pulls toward the more precise 100.05 m value
Best estimate of a quantity from observations with unequal reliability; gives higher influence to more precise measurements.
Diagrams to know
- Weighted mean visualization showing contribution of each observation
- Weight distribution pie chart for typical network
Practical Leveling & Distance Measurement Applications
Formulas
wᵢ = 1/Kᵢ (relative weights for leveling)
Meaning
wᵢ = relative weight; Kᵢ = distance of leveling route i in km
Watch out
Use DISTANCE (path length), not elevation difference. Longer loops accumulate more random error.
When to use
In spirit or trigonometric leveling networks, assign weights inversely to distance.
σleveling ≈ c·√K
Meaning
σ = standard deviation of level network; c = constant (depends on method); K = total route distance in km
Watch out
Error grows as √K, NOT linearly with K. Doubling network length increases error by √2 ≈ 1.41, not 2.
When to use
Estimate expected precision of leveling given total network distance; standard deviation grows with √K.
Important facts
- In leveling networks, shorter routes are inherently more precise due to fewer setup stations and less accumulated error.
- Weight assignment: if three routes are 2 km, 1 km, and 4 km, relative weights are 1/2, 1/1, 1/4 = 0.5, 1.0, 0.25.
- Standard deviation in leveling roughly proportional to √(distance); longer networks naturally have lower precision.
- For Philippine leveling surveys (RA 8560), network design should account for distance-based weight distribution.
- PRS92 datum implementation often uses multiple leveling routes; proper weighting is essential for adjustment.
Key definitions
Leveling Route Distance (K)
Example
A closed leveling loop from point A → B → C → A covering 2 + 1.5 + 1.5 = 5 km has K = 5 km
Total horizontal distance (in km) traveled along a leveling circuit; primary factor determining measurement precision.
Diagrams to know
- Leveling network diagram showing routes with distances and assigned weights
- Weight reduction curve: w vs. distance (hyperbolic decay)
Numerical Worked Examples (Board-Exam Style)
Important facts
- EXAMPLE 1 — Arithmetic mean and standard deviation: Five distance measurements are 100.01, 100.02, 100.03, 100.02, 100.02 m. Find mean and std dev. SOLUTION: x̄ = 500.10/5 = 100.02 m. Residuals: v = [−0.01, 0, +0.01, 0, 0]. Σv² = 0.0002. σ = √(0.0002/4) = √0.00005 = 0.00707 m ≈ 0.007 m.
- EXAMPLE 2 — Precision improvement with replication: A measurement has σ = 0.024 m from a single observation. If measured 9 times, what is the standard deviation of the mean? SOLUTION: σₓ̄ = 0.024/√9 = 0.024/3 = 0.008 m. Precision improves by factor 3.
- EXAMPLE 3 — Weighted mean with two observations: Distance measured twice: 100.02 m (σ = 0.02 m) and 100.05 m (σ = 0.01 m). Find weighted mean. SOLUTION: w₁ = 1/0.02² = 2,500; w₂ = 1/0.01² = 10,000. x̄w = (2,500 × 100.02 + 10,000 × 100.05) / 12,500 = (250,050 + 1,000,500) / 12,500 = 1,250,550 / 12,500 = 100.0440 m.
- EXAMPLE 4 — Leveling weights by distance: Three level routes to a bench mark have lengths 3 km, 5 km, and 2 km. Assign relative weights. SOLUTION: w ∝ 1/K. w₁ = 1/3 ≈ 0.333; w₂ = 1/5 = 0.2; w₃ = 1/2 = 0.5. Normalize to integers: multiply by 6 → weights are 2, 1.2, 3. Or keep as 1/3, 1/5, 1/2.
- EXAMPLE 5 — Standard deviation of weighted mean: Three observations with values 50.10, 50.14, 50.12 and weights 2, 3, 1. Find weighted mean and std dev of mean. SOLUTION: x̄w = (2×50.10 + 3×50.14 + 1×50.12)/(2+3+1) = (100.20 + 150.42 + 50.12)/6 = 300.74/6 = 50.1233 m. Residuals: v₁ = −0.0233, v₂ = +0.0167, v₃ = −0.0033. σw = √[(2×0.0233² + 3×0.0167² + 1×0.0033²)/(3−1)] = √[0.00109]/2 ≈ 0.0233. σx̄w = 0.0233/√6 ≈ 0.0095 m.
Must remember
- Weight w = 1/σ² (INVERSE of VARIANCE, not std dev). Never use 1/σ.
- Arithmetic mean x̄ = Σx/n is the MPV for EQUAL-weight observations only.
- Weighted mean x̄w = Σ(wᵢ·xᵢ)/Σwᵢ for UNEQUAL-weight observations (foundation of least squares).
- Standard deviation σ = √(Σv²/(n−1))—use n−1, NOT n. Sample std dev, not population.
- Precision improves by √n, NOT n. Adding 4 measurements reduces σₓ̄ by factor 2, not 4.
- Leveling weight w ∝ 1/K (inverse distance); shorter route → higher weight → more reliable.
- Residual sum: Σv = 0 for unweighted; Σ(wᵢ·vᵢ) ≈ 0 for weighted (internal check).
- Standard deviation of mean: σₓ̄ = σ/√n (unweighted); σx̄w = σw/√(Σwᵢ) (weighted—use weight sum, not count).
- Eliminate BLUNDERS and CORRECT SYSTEMATIC errors BEFORE adjustment; only RANDOM errors are handled by least squares.
- Probable error PE = 0.6745σ—represents 50% confidence interval; memorize the constant 0.6745.
Last minute tips
- Always double-check: Is the problem asking for a SINGLE-OBSERVATION std dev (σ) or the MEAN std dev (σₓ̄)? They differ by √n. Wrong choice = wrong answer.
- For leveling problems: Weight w ∝ 1/DISTANCE. Shorter route = higher weight. Verify by logic: 1 km route should be more reliable (higher weight) than 5 km route.
- Verify Σv ≈ 0 (unweighted) or Σ(wᵢ·vᵢ) ≈ 0 (weighted) after computing residuals—catches arithmetic errors before you finish.
- Remember: n−1 in σ formula, not n. Using n is a common board exam trap and underestimates error significantly.
- If given variance σ², convert to std dev first (σ = √(σ²)), then compute weights. Forgetting this step is a frequent mistake.
Comparison tables
Error Types: How to Identify & Handle
| Error Type | Magnitude | Sign Pattern | Cause | How to Handle | Use in Adjustment? |
|---|---|---|---|---|---|
| Blunder | Large (>3σ) | Random | Misreading, instrument failure | Detect & eliminate before adjustment | NO—reject observation |
| Systematic | Constant or gradual | Always same sign | Temperature, equipment bias | Model mathematically & correct | NO—correct first, then adjust |
| Random | Small (<3σ) | Mixed +/− | Inherent instrument/observer limit | Statistical adjustment (least squares) | YES—adjust using weights & MPV |
Unweighted vs. Weighted Mean Comparison
| Aspect | Unweighted Mean | Weighted Mean |
|---|---|---|
| Formula | x̄ = Σx/n | x̄w = Σ(wᵢ·xᵢ)/Σwᵢ |
| When to use | All observations equal precision | Observations have unequal precision |
| Weight basis | All equal (implicitly w = 1) | w = 1/σ² (explicit assignment) |
| Residual sum | Σv = 0 (always) | Σ(wᵢ·vᵢ) ≈ 0 (as check) |
| Std dev of mean | σₓ̄ = σ/√n | σx̄w = σw/√(Σwᵢ) |
| Influence of outlier | Equal (full 1/n share) | Reduced (low weight if poor precision) |
Weight Assignment Rules Summary
| Situation | Weight Formula | Proportionality | Example |
|---|---|---|---|
| From variance/precision | w = 1/σ² | w ∝ 1/σ² (inverse) | σ = 0.01 m → w = 10,000; σ = 0.02 m → w = 2,500 |
| Leveling by distance | w = 1/K (relative) | w ∝ 1/K (inverse) | K = 1 km → w = 1; K = 2 km → w = 0.5 |
| Repeated measurements | w = n (relative) | w ∝ n (direct) | n = 3 repeats → w = 3; n = 1 → w = 1 |
| Combined rules | w = n/σ² (or n/K) | w ∝ n; w ∝ 1/σ² | If obs repeated 4 times with σ = 0.02: w = 4/0.0004 = 10,000 |
Standard Deviation Formulas: Single Observation vs. Mean
| Statistic | Unweighted Formula | Weighted Formula | Interpretation | Board Exam Pitfall |
|---|---|---|---|---|
| Single observation | σ = √(Σv²/(n−1)) | σw = √(Σ(wᵢ·vᵢ²)/(n−1)) | Scatter of individual measurements | Use n−1, NOT n. One DoF lost fitting mean. |
| Mean/MPV | σₓ̄ = σ/√n | σx̄w = σw/√(Σwᵢ) | Precision of final answer | Denominator in weighted case is √(Σwᵢ), not √n. Weight sum matters. |
| Improvement factor | √n (always) | √(Σwᵢ) (for equal wᵢ = √n) | More measurements → √n× better precision | Not linear! Doubling n improves by √2 ≈ 1.41, not 2. |
Diagrams
Error Theory & Weight Framework
Hierarchical overview of error theory: classification pathways, MPV selection logic, weight rules, and standard deviation contexts. Shows decision tree for choosing unweighted vs. weighted approaches.
Adjustment Decision & Formula Selection
Step-by-step flowchart for observation adjustment: decision point (equal vs. unequal weights), residual computation, verification checks (Σv=0), and standard deviation selection. Highlights common pitfalls and verification gates.
Observation States: Blunder Detection to Adjustment
State diagram showing observation flow: raw data → error classification → elimination/correction/adjustment. Illustrates proper sequence: eliminate blunders, correct systematic errors, then apply statistical adjustment to random errors.
Example Leveling Network Weights
Typical leveling network with three routes: 2 km (weight 50%), 1 km (weight 100%, highest), 4 km (weight 25%). Demonstrates inverse proportionality to distance: shortest route dominates weighted mean.