Measurements and Theory of Errors - Cheat Sheet
Your last-minute revision companion for PRC Board Exam: rapid-fire formulas, definitions, and exam traps for tape corrections, error propagation, and most probable value.
Error Classification & Definitions
Common values
Value
0.6745
Symbol
k
Quantity
Probable Error coefficient
Value
σ ÷ E = 1.4826
Symbol
ratio
Quantity
Standard deviation to probable error
Important facts
- Mistakes are not random; eliminate them completely by re-measurement and verification.
- Systematic errors accumulate linearly; apply each correction separately.
- Random errors accumulate in quadrature (√ of sum of squares).
- More measurements → smaller probable error of the mean: E_m = E / √n.
- Statistical treatment assumes random errors follow normal distribution (Gaussian curve).
- Systematic errors must be corrected before statistical analysis.
- A tape that is too long causes measured distances to read short.
- All tape corrections are additive (algebraic sum).
Key definitions
Mistake (Blunder)
Example
Reading 150 m instead of 15.0 m; recording wrong station number.
Gross human error (misreading, wrong entry); must be eliminated; no statistical treatment.
Systematic Error
Example
Tape expanding 0.02 m per 10 °C rise; sag shortening each span.
Follows a physical law; repeatable and correctable; cause is identifiable (temperature, sag, tension, instrument calibration).
Random (Accidental) Error
Example
±0.01 m variation in tape reading; slight parallax in level sighting.
Small, unavoidable variations from causes too numerous/complex to control; treated statistically; approximately normal distribution.
Most Probable Value (MPV)
Example
Three readings: 100.1, 100.2, 100.0 m → MPV = 100.1 m.
The arithmetic mean of repeated measurements; best estimate when random errors are present.
Probable Error (E)
Example
E = ±0.03 m means 50% chance true value lies within ±0.03 m of a single reading.
Half-width of the interval containing 50% of all observations; measure of scatter in single measurement.
Residual (v)
Example
If MPV = 100.1 m and x = 100.2 m, then v = 0.1 m.
Difference between a measured value and the most probable value: v = x − MPV.
Diagrams to know
- Normal (Gaussian) distribution curve showing probable error (±0.6745σ)
- Tape correction sign convention (+ lengthens, − shortens)
- Error propagation flowchart (which errors combine how)
Most Probable Value & Error of the Mean
Formulas
MPV = x̄ = (Σx) / n
Meaning
x = measured values; n = number of measurements; Σ = sum
Watch out
Do not include obvious mistakes; remove blunders first. Do not forget units (must match all measurements).
When to use
Always: to find best estimate from repeated observations.
E = 0.6745 × √(Σv² / (n−1))
Meaning
E = probable error of single observation; v = residual (x − MPV); n = count; 0.6745 is standard coefficient.
Watch out
Use n−1, not n, in denominator (Bessel correction). Do not confuse this E with elasticity E in tension correction.
When to use
To quantify scatter in repeated measurements; for n ≥ 4 use n−1 (sample correction).
E_m = E / √n
Meaning
E_m = probable error of the mean; E = probable error of single measurement; n = number of measurements.
Watch out
E_m < E always. Doubling n reduces E_m by √2 ≈ 1.41, not by 2.
When to use
To find uncertainty in the final answer (MPV). More measurements → smaller E_m.
Common values
Value
1.414
Symbol
—
Quantity
√2
Value
1.732
Symbol
—
Quantity
√3
Value
2.0
Symbol
—
Quantity
√4
Value
2.236
Symbol
—
Quantity
√5
Important facts
- Mean always has smaller error than any single measurement.
- Error decreases with √n, so diminishing returns beyond ~10 readings.
- Residuals must sum to (approximately) zero: Σv ≈ 0 (property of mean).
- Probable error is not the same as standard deviation (related by factor 0.6745).
- For 3 measurements, √n ≈ 1.73; E_m ≈ 0.58E.
Key definitions
Sample Standard Deviation (σ̂)
Example
If Σv² / (n−1) = 0.0025, then σ̂ = 0.05 m; E = 0.034 m.
σ̂ = √(Σv² / (n−1)); related to E by E = 0.6745σ̂.
Diagrams to know
- Bell curve showing single measurement E and mean E_m
- Table: n vs √n showing how error of mean decreases
Tape (Distance) Corrections
Formulas
C_t = α L (T − T_s)
Meaning
C_t = temperature correction; α = linear expansion coefficient (steel: 11.6×10⁻⁶/°C); L = measured length; T = field temp; T_s = standard temp (usually 20°C).
Watch out
α has units °C⁻¹; if T_s = 20°C and T = 30°C, then (T − T_s) = +10°C. Steel coeff ≈ 11.6, invar ≈ 0.9×10⁻⁶/°C (invar negligible). Always multiply: C_t = α × L × ΔT.
When to use
Every measurement with tape; add if T > T_s (tape expands, so correction is positive, making true length larger).
C_p = (P − P_s) L / (A E)
Meaning
C_p = tension correction; P = field pull (N); P_s = standard pull (usually 45 N); L = measured length; A = cross-sectional area (mm²); E = Young's modulus (for steel ~200 GPa = 200,000 N/mm²).
Watch out
This E is Young's modulus, NOT probable error. Use consistent units: if A in mm² and E in N/mm², result in meters. C_p is usually tiny (< 0.01 m for typical 30 m tape).
When to use
When field pull differs from standard (e.g., pulling harder on slope); add if P > P_s (more tension → longer tape reading).
C_sag = −W² L / (24 P²)
Meaning
C_sag = sag correction (always negative, shortens); W = total weight of suspended portion (N); L = measured length; P = tension (pull, N).
Watch out
Always negative. Formula uses W = total weight (not weight per unit length w). Alternative: C_sag = −w² L³ / (24 P²) if w = weight per unit length (N/m); both forms equivalent.
When to use
When tape is unsupported between supports (usually only ends); correction is always subtracted (sag makes tape shorter than straight line).
C_sag (alt) = −w² L³ / (24 P²)
Meaning
Alternative: w = weight per unit length (N/m); L³ appears instead of L when using w.
Watch out
Must know which form: W (total) or w (per unit). Check units: w in N/m gives C_sag in meters only if dimensional analysis holds.
When to use
Same condition as above; use whichever form is given or convenient.
H = √(L² − h²)
Meaning
H = horizontal distance; L = slope distance; h = elevation difference between ends.
Watch out
h must be measured perpendicular to ground (vertical). For small h (h << L), approximate: H ≈ L − h²/(2L).
When to use
Converting slope measurement to horizontal (most common in terrain with relief).
C_h = −h² / (2L)
Meaning
C_h = slope correction (always negative, horizontal < slope); h = elevation difference; L = slope distance.
Watch out
This is an approximation; max error ~0.0005 m per 100 m for h/L < 0.1. Always negative (horizontal is shorter).
When to use
Quick approximation when h is small (< 10 m per 100 m tape); exact formula is √(L² − h²).
True distance = Measured × (Actual tape length / Nominal tape length)
Meaning
If tape is too long: actual > nominal → multiply by factor > 1 → true distance > measured. If too short: actual < nominal → multiply by factor < 1.
Watch out
Tape too long → measured reads SHORT → multiply by ratio > 1 to get true. Inverse rule for laying out: if setting a distance with a long tape, lay out a smaller measurement.
When to use
When the tape itself is not exactly 30 m (or whatever length stamped on it); found by laying tape against a calibrated baseline.
Common values
Value
11.6 × 10⁻⁶
Symbol
α (per °C)
Quantity
Linear expansion, steel
Value
0.9 × 10⁻⁶
Symbol
α (per °C)
Quantity
Linear expansion, invar
Value
200 GPa = 200,000 N/mm²
Symbol
E
Quantity
Young's modulus, steel
Value
45 N (sometimes 50 N)
Symbol
P_s
Quantity
Standard pull (tension)
Value
20°C (sometimes 68°F ≈ 20°C)
Symbol
T_s
Quantity
Standard temperature
Important facts
- Apply each correction separately; sum algebraically.
- Temperature & tension corrections can be positive or negative.
- Sag correction is always negative.
- Slope correction is always negative.
- For gentle slopes, use √(L² − h²); for steep slopes or high precision, avoid sag by supporting entire tape.
- Typical correction magnitudes: temp ±0.01–0.05 m/100 m; sag −0.001–0.01 m/span; tension ±0.001 m; slope −0.001–0.05 m depending on angle.
- Tape calibration error (actual ≠ nominal) is cumulative, not per-span.
Key definitions
Standard Conditions
Example
Field: 30°C, 60 N, spans unsupported → three corrections needed.
Reference state for tape: usually 20°C and 45 N pull, supported throughout; any deviation requires correction.
Sag
Example
30 m tape pulled with 50 N sag = ~0.05 m for 10 m unsupported span.
Vertical dip of unsupported tape span; caused by self-weight; measured as perpendicular distance from chord to tape.
Diagrams to know
- Sign convention for all four tape corrections (which are +, which are −)
- Slope to horizontal: right triangle showing L, h, H
- Sag curve: tape suspended between points, showing sag distance
Error Propagation
Formulas
E_sum = √(E₁² + E₂² + E₃² + ... + Eₙ²)
Meaning
E_sum = probable error of a sum; E₁, E₂, ..., Eₙ = probable errors of each measured quantity.
Watch out
Errors combine in quadrature (square-root of sum of squares), NOT linearly. If all errors equal (E each), then E_sum = E√n.
When to use
When adding or subtracting independent measurements (e.g., summing traverse legs, height differences in leveling).
E_sum = E√n (special case: n equal errors)
Meaning
If each of n measurements has the same probable error E, the sum has error E√n.
Watch out
Error of sum grows with √n, not linearly. Error of mean shrinks with √n. Do not confuse them.
When to use
Common in repeated operations: e.g., 10 tape lengths, each ±0.02 m → E_sum = 0.02√10 ≈ 0.063 m.
E_product (relative) ≈ √((E₁/L₁)² + (E₂/L₂)² + ...)
Meaning
For a product or quotient, the relative error (not absolute) combines in quadrature; then multiply by the result to get absolute error.
Watch out
This is for relative errors; convert to absolute at the end: E_product = (result) × (relative error).
When to use
Area = L × W; if errors are E_L and E_W, then relative error in area is √((E_L/L)² + (E_W/W)²).
Important facts
- Random errors partially cancel when combined in quadrature.
- Quadrature sum is always < linear sum (except for single measurement).
- Systematic errors combine linearly; do not use √(Σ²) for systematic errors.
- Error of traverse sum: √(n × E_single²) where n = number of legs.
- Relative error in product: √((δA/A)² + (δB/B)²) for C = A·B.
- For division: same rule as product (relative errors).
- In level runs: vertical error accumulates as √(number of setups) × (per-setup error).
Key definitions
Error of the Sum
Example
Traverse of 5 legs, each ±0.01 m → total error ≈ ±0.022 m.
Probable error in a total distance or height found by adding independent measurements; always larger than any single error.
Quadrature (Root Sum Squares)
Example
E₁ = 0.03, E₂ = 0.04 → E_sum = √(0.03² + 0.04²) = 0.05, not 0.07.
Method of combining independent random errors: E_total = √(ΣE_i²); accounts for random cancellation.
Diagrams to know
- Quadrature vs linear combination (graph showing why √ not arithmetic sum)
- Flowchart: how to choose between quadrature and linear propagation
Common Board-Exam Traps & Rules
Important facts
- Measuring with long tape: true = measured × (actual/nominal), result > measured.
- Laying out with long tape: lay = desired × (nominal/actual), lay < desired.
- Sign convention: all corrections are added; negative result means subtract.
- Sag always shortens — it's a natural consequence of gravity and cable theory.
- Slope always shortens — horizontal is the shortest path between two points at different elevations.
- Temperature increases → tape expands → measured distances read shorter (when using a tape that is longer than nominal).
- Higher pull (tension) → measured distances read longer (tape is stretched).
- Sag effect is huge if tape is unsupported: ~0.05 m per 10 m unsupported span with typical pull.
- Order of corrections doesn't matter (algebra is commutative); total = C_t + C_p + C_sag + C_h.
- Do NOT apply slope correction if using √(L² − h²) directly — it's already correct; use C_h only as an approximation.
Key definitions
Tape Too Long vs Tape Too Short
Example
Tape reads 30.00 m but is actually 30.02 m. Measuring 150 m → true = 150 × (30.02/30) = 150.1 m. Laying out 150 m → set 150 × (30/30.02) = 149.9 m on tape.
Too long: actual length > nominal (stamped) length. When used to measure, each 30 m segment is actually longer, so the true distance is greater than the measured reading. When used to lay out, each segment must be set shorter to compensate.
Must remember
- Probable error of single measurement: E = 0.6745√(Σv²/(n−1)) — use n−1 (Bessel correction); not the same as standard deviation.
- Error of mean: E_m = E/√n — more readings → exponentially smaller error in final answer; diminishing returns after ~10–15 readings.
- Tape too long → measured reads SHORT → true = measured × (actual/nominal) — inverse rule when laying out.
- All four tape corrections combine algebraically: Corrected = Measured + C_t + C_p + C_sag + C_h — sag and slope are always negative.
- Errors combine in quadrature: E_sum = √(E₁² + E₂² + ...), NOT linear addition — random errors partially cancel; quadrature sum < arithmetic sum.
- Systematic errors (temperature, tension, sag, slope) must be corrected BEFORE statistical analysis — random errors are treated after.
- Sag correction always shortens the tape: C_sag = −W²L/(24P²) — heavier tape or lower tension → larger sag effect.
- Slope correction always shortens the distance: C_h = −h²/(2L) — horizontal is the shortest path; exact: H = √(L² − h²).
- Temperature coefficient of steel tape: α ≈ 11.6 × 10⁻⁶ /°C; invar ≈ 0.9 × 10⁻⁶ /°C — invar is preferred for high precision.
- Distinguish between blunders (eliminate), systematic (correct), and random (average/statistics) — each requires a different action.
Last minute tips
- In a multiple-choice exam on tape corrections: always check the sign (+ or −) first. If tape is long, measured < true; if temperature rose, tape expanded (measure short with long tape). Draw a quick diagram.
- For 'probable error' calculations, always use n−1 in the denominator (sample standard deviation), not n. This is the Bessel correction — most common exam trick.
- When asked 'error of the sum' in a traverse or leveling run, use √(Σ E_i²), not simple addition. If you see a numerical answer that looks too small, you probably forgot the square root.
- Tape corrections are independent — apply each one. For a 30 m tape at 35°C with 60 N pull over sloped terrain, you need all four corrections (temperature, tension, slope, sag if unsupported).
- In error propagation, always ask: 'Are these errors independent?' If yes → quadrature. If no (e.g., same tape used for all legs) → treat partially as systematic. Read the problem carefully.
Comparison tables
Systematic vs Random Errors
| Characteristic | Systematic | Random (Accidental) |
|---|---|---|
| Cause | Physical law; identifiable | Unknown; too complex to control |
| Pattern | Repeatable; follows same direction/magnitude | No pattern; varies unpredictably |
| Correction | Apply mathematical correction; subtract from measurement | Cannot correct; only reduce by improved technique or averaging |
| Treatment | Before statistical analysis | After correction, use statistics |
| Example | Tape too long; temperature expansion; sag | Reading error; vibration; minor centering error |
| Combination | Arithmetic (linear) sum | Quadrature (root-sum-square) |
Tape Corrections Summary
| Correction | Formula | Sign | Physical Reason | Typical Magnitude (30 m tape) |
|---|---|---|---|---|
| Temperature | C_t = α L (T − T_s) | + if T > T_s | Tape expands/contracts | ±0.03 m per 10°C |
| Tension/Pull | C_p = (P − P_s)L / (AE) | + if P > P_s | Tape stretches under load | ±0.001 m per 10 N |
| Sag | C_sag = −W²L / (24P²) | Always − | Gravity pulls tape down | −0.001 to −0.05 m per span |
| Slope | C_h = −h² / (2L) | Always − | Horizontal < slope distance | −0.0001 to −0.01 m per 30 m |
When to Use Each Error Propagation Rule
| Scenario | Formula | Example | Result is Larger/Smaller? |
|---|---|---|---|
| Sum of independent measurements with different errors | E_sum = √(E₁² + E₂² + ...) | Traverse of 3 legs: ±0.02, ±0.03, ±0.01 m | Quadrature; smaller than linear sum |
| Sum of n identical independent measurements | E_sum = E√n | 10 leveling setups, each ±0.005 m | Grows with √n |
| Mean of n measurements | E_mean = E / √n | 10 distance readings, single error ±0.02 m | Shrinks with √n |
| Product or quotient of measurements | Relative error = √((E_A/A)² + (E_B/B)²) | Area = L × W, then E_area = area × rel_error | Depends on relative errors |
Diagrams
Tape Correction Decision Tree
Step-by-step decision logic for applying tape corrections. Follow each branch to identify which corrections apply, then sum algebraically.
Measurements & Errors — Concept Map
Comprehensive mind map showing the relationships between error types, MPV calculations, tape corrections, and error propagation formulas. Central concept branches into four major topics.
Error Propagation Formula Selection
Decision flowchart for selecting the correct error propagation formula. Branches on operation type and whether errors are identical or different.