Tags: sample mean, standard deviation, uncertainty, error propagation, partial derivatives, volume uncertainty, caliper measurements, best estimate, ENGR 216, PHYS 216, Texas A&M
Every measurement has uncertainty, and the job is to quantify it honestly. You calculate uncertainty in a single measurement set using the sample standard deviation divided by the square root of the number of readings. When a derived quantity depends on several measured variables, you propagate their individual uncertainties through partial derivatives.
Sample mean (x̄)
The arithmetic average of a set of N measurements. Sum all values and divide by N.
Sample standard deviation (s)
A measure of how spread out individual measurements are around the mean. Calculated using the (N − 1) denominator (Bessel's correction) to avoid underestimating spread from a finite sample.
Uncertainty of the best estimate (σ_x̄)
Also called the standard error of the mean. Equal to the sample standard deviation divided by √N. This tells you how confident you are in the mean itself, not in any single reading.
Error propagation (uncertainty propagation)
The method for finding the uncertainty in a calculated quantity when that quantity is a function of one or more measured variables, each with its own uncertainty.
Partial derivative
The derivative of a multivariable function with respect to one variable while holding the others constant. Central to propagating uncertainty through formulas.
Add all measured values together, then divide by the number of measurements N.
Formula: x̄ = (1/N) Σ xᵢ
The mean is your "best estimate" of the true value.
Worked example (Problem 1):
Given x₁ = a + 3, x₂ = a, x₃ = a − 4, x₄ = a + 5, x₅ = a + 2:
Sum = 5a + (3 + 0 − 4 + 5 + 2) = 5a + 6
Mean = (5a + 6) / 5 = a + 1.2
Answer: a + 1.2
Start with the sample standard deviation s.
Divide by √N to get the standard error: σ_x̄ = s / √N
This shrinks as you take more measurements, which makes sense: more data, more confidence.
Worked example (Problem 2):
Given: 10 measurements, sample mean = 62.7, sample standard deviation s = 2.41.
σ_x̄ = 2.41 / √10 = 2.41 / 3.1623 = 0.7622...
Rounded to two decimal places: 0.76
When a quantity is the product (or quotient) of measured values, the fractional (relative) uncertainties add in quadrature:
For V = L × W × H:
δV/V = √[(δL/L)² + (δW/W)² + (δH/H)²]
Worked example (Problem 3):
Rectangular block sides: L = 25.0 ± 0.08 mm, W = 12.5 ± 0.03 mm, H = 6.0 ± 0.02 mm.
V = 25.0 × 12.5 × 6.0 = 1875.0 mm³
Relative uncertainties:
δL/L = 0.08 / 25.0 = 0.0032
δW/W = 0.03 / 12.5 = 0.0024
δH/H = 0.02 / 6.0 = 0.00333
Combined relative uncertainty = √(0.0032² + 0.0024² + 0.00333²)
= √(0.00001024 + 0.00000576 + 0.0000111)
= √(0.0000271) = 0.005206
δV = 1875.0 × 0.005206 ≈ 9.76 mm³
The volume is reported as 1875 ± 10 mm³ (rounding the uncertainty to match significance).
For any function Q = f(x, y, z, ...) with independent variables:
δQ = √[(∂Q/∂x · δx)² + (∂Q/∂y · δy)² + (∂Q/∂z · δz)² + ...]
Each term is: the partial derivative of Q with respect to that variable, multiplied by the uncertainty in that variable. Then combine all terms in quadrature (root-sum-of-squares).
Worked example (Problem 4):
Q = (x + 2) / [x + y·cos(4θ)], with x = 10 ± 2, y = 7 ± 1, θ = 40° ± 3°.
Step 1: Convert θ to radians for calculus. θ = 40° = 0.6981 rad, δθ = 3° = 0.05236 rad.
Step 2: Evaluate Q at central values.
cos(4 × 40°) = cos(160°) = −0.9397
Denominator = 10 + 7(−0.9397) = 10 − 6.578 = 3.422
Q = 12 / 3.422 = 3.506
Step 3: Compute partial derivatives.
∂Q/∂x = [1·(x + y cos4θ) − (x+2)·1] / (x + y cos4θ)² = (y cos4θ − 2) / (denom)²
= (−6.578 − 2) / (3.422)² = −8.578 / 11.71 = −0.7325
∂Q/∂y = −(x+2)·cos(4θ) / (denom)² = −12·(−0.9397) / 11.71 = 11.276 / 11.71 = 0.9629
∂Q/∂θ = (x+2)·4y·sin(4θ) / (denom)² = 12 × 4 × 7 × sin(160°) / 11.71
sin(160°) = 0.3420
= 12 × 28 × 0.3420 / 11.71 = 114.91 / 11.71 = 9.814
Step 4: Combine.
δQ = √[(−0.7325 × 2)² + (0.9629 × 1)² + (9.814 × 0.05236)²]
= √[(−1.465)² + (0.9629)² + (0.5143)²]
= √[2.146 + 0.9272 + 0.2645]
= √3.338 ≈ 1.83
Worked example (Problem 5):
Q = (1 − x²)·cos[(x + 2)/x³], with x = 1.70 ± 0.02.
Since Q depends on only one variable, δQ = |dQ/dx| × δx.
This derivative is found by applying the product rule and chain rule. At x = 1.70:
(x + 2)/x³ = 3.70 / 4.913 = 0.7531
cos(0.7531) = 0.7288
1 − x² = 1 − 2.89 = −1.89
Q = −1.89 × 0.7288 = −1.377
For the derivative dQ/dx, use the product rule on (1 − x²) and cos[(x+2)/x³]:
d/dx(1 − x²) = −2x = −3.40
Let u = (x+2)/x³. Then du/dx = [x³ − 3x²(x+2)] / x⁶ = [x − 3(x+2)] / x⁴ = (−2x − 6) / x⁴
At x = 1.70: du/dx = (−3.40 − 6) / 8.3521 = −9.40 / 8.3521 = −1.1254
d/dx[cos(u)] = −sin(u) · du/dx = −sin(0.7531) × (−1.1254) = 0.6847 × 1.1254 = 0.7706
Product rule: dQ/dx = (−3.40)(0.7288) + (−1.89)(0.7706) = −2.478 − 1.456 = −3.934
δQ = |−3.934| × 0.02 ≈ 0.079
Sample mean: x̄ = (1/N) Σ xᵢ
Standard error of the mean: σ_x̄ = s / √N
Propagation for products/quotients (relative form): δQ/Q = √[Σ (δxᵢ/xᵢ)²]
General propagation (absolute form): δQ = √[Σ (∂Q/∂xᵢ × δxᵢ)²]
⚠️ The distinction between standard deviation (spread of data) and standard error (uncertainty in the mean) is a classic exam trap. Know which one the question is asking for.
⚠️ When propagating uncertainty through trig functions, always convert angle uncertainties to radians before multiplying by the partial derivative.
⚠️ For products and quotients, use relative (fractional) uncertainties. For sums and differences, use absolute uncertainties directly.
⚠️ Partial derivatives in the propagation formula are evaluated at the central (measured) values, not at the uncertainty bounds.
Q: What is the difference between sample standard deviation and the uncertainty of the best estimate?
A: Sample standard deviation (s) describes how spread out individual measurements are. The uncertainty of the best estimate (σ_x̄ = s/√N) describes how well you know the mean. As N increases, the mean becomes more certain even if individual readings remain scattered.
Q: You measure a cylinder's radius as 5.0 ± 0.1 cm and its height as 12.0 ± 0.2 cm. What is the relative uncertainty in the volume (V = πr²h)?
A: Since V = π·r²·h, and π is exact: δV/V = √[(2·δr/r)² + (δh/h)²] = √[(2×0.1/5.0)² + (0.2/12.0)²] = √[(0.04)² + (0.01667)²] = √[0.0016 + 0.000278] = √0.001878 ≈ 0.0433, so about 4.3%.
Q: Why do we use (N − 1) in the denominator of sample standard deviation instead of N?
A: Because we estimated the mean from the same data. This "uses up" one degree of freedom, so dividing by (N − 1) corrects for the slight underestimate that dividing by N would produce. This is Bessel's correction.
sample mean, arithmetic mean, standard deviation, Bessel's correction, standard error of the mean, uncertainty of the best estimate, error propagation, uncertainty propagation, partial derivatives, quadrature, root sum of squares, RSS, relative uncertainty, fractional uncertainty, absolute uncertainty, caliper measurement, volume uncertainty, ENGR 216, PHYS 216, Texas A&M, experimental physics