Source: ENGR/PHYS 216 Final Exam Practice Problems
Tags: sample mean, standard deviation, uncertainty, error propagation, partial derivatives, best estimate, ENGR 216, Texas A&M
These problems cover the statistical foundations of experimental measurement: computing sample means, quantifying uncertainty from repeated measurements, and propagating uncertainty through formulas using partial derivatives. Nearly every ENGR 216 exam will include at least one error propagation problem, so the partial-derivative method is essential.
Sample mean (x̄)
The arithmetic average of a set of n measurements. Calculated as x̄ = (1/n) Σxᵢ. This is the "best estimate" of the true value.
Sample standard deviation (s)
A measure of the spread in individual measurements around the mean. Uses n − 1 in the denominator (Bessel's correction): s = √[ Σ(xᵢ − x̄)² / (n − 1) ]
Uncertainty of the best estimate (standard error)
The uncertainty in the sample mean itself, not in individual measurements. Calculated as: δx̄ = s / √n
This shrinks as you take more measurements, because averaging reduces noise.
Error propagation (uncertainty propagation)
The method for finding the uncertainty in a calculated quantity Q = f(x, y, z, ...) when x, y, z each have their own uncertainties. For independent, random errors: δQ = √[ (∂Q/∂x · δx)² + (∂Q/∂y · δy)² + (∂Q/∂z · δz)² + ... ]
Partial derivative (∂Q/∂x)
The rate of change of Q with respect to one variable while holding all others constant. Each partial derivative acts as a "sensitivity coefficient" telling you how much Q changes per unit change in that variable.
The sample mean is a straightforward sum-and-divide operation. When the data is given in terms of a parameter (like a), the parameter carries through the algebra.
Sum all values, divide by the number of measurements
Symbolic parameters remain in the result
Worked example (Problem 1): Given x₁ = a + 3, x₂ = a, x₃ = a − 4, x₄ = a + 5, x₅ = a + 2:
x̄ = (5a + 3 + 0 − 4 + 5 + 2) / 5 = (5a + 6) / 5 = a + 1.2
When you have n repeated measurements, the uncertainty in your best estimate (the mean) is the standard error:
δx̄ = s / √n
This is distinct from the standard deviation s, which describes the scatter of individual data points. The standard error describes how confident you are in the mean.
Worked example (Problem 2): Ten measurements with x̄ = 62.7 and s = 2.41:
δx̄ = 2.41 / √10 = 2.41 / 3.162 = 0.76
For a quantity that is a product/quotient of measured values, the relative uncertainties add in quadrature. For V = L × W × H:
δV / V = √[ (δL/L)² + (δW/W)² + (δH/H)² ]
Then multiply both sides by V to get the absolute uncertainty δV.
Worked example (Problem 3): Rectangular block with sides 25.0 ± 0.08 mm, 12.5 ± 0.03 mm, 6.0 ± 0.02 mm. V = 1875.0 mm³.
δV = 1875.0 × √[(0.08/25.0)² + (0.03/12.5)² + (0.02/6.0)²] δV = 1875.0 × √[1.024×10⁻⁵ + 5.76×10⁻⁶ + 1.111×10⁻⁵] δV = 1875.0 × 0.005207 δV = 9.76 mm³
For any function Q = f(x, y, θ, ...), compute each partial derivative, multiply by the corresponding uncertainty, then combine in quadrature.
⚠️ If a variable is in degrees, convert its uncertainty to radians before plugging in. The partial derivative with respect to an angle is naturally in radians.
Worked example (Problem 4): Q = (x + 2) / (x + y · cos(4θ)), with x = 10 ± 2, y = 7 ± 1, θ = 40° ± 3°.
Step 1 – Evaluate Q: cos(160°) = −0.9397 Q = 12 / (10 + 7 × (−0.9397)) = 12 / 3.422 = 3.507
Step 2 – Compute partial derivatives (numerically or analytically): ∂Q/∂x = −0.732, ∂Q/∂y = 0.963, ∂Q/∂θ = 9.813 (in rad⁻¹)
Step 3 – Convert δθ to radians: 3° × π/180 = 0.05236 rad
Step 4 – Combine: δQ = √[(−0.732 × 2)² + (0.963 × 1)² + (9.813 × 0.05236)²] δQ = √[2.146 + 0.927 + 0.264] δQ = 1.83
When Q depends on only one variable, the formula simplifies to:
δQ = |dQ/dx| × δx
The derivative can be computed analytically or numerically (finite difference method).
Worked example (Problem 5): Q = (1 − x²) · cos((x + 2) / x³), with x = 1.70 ± 0.02.
Using numerical differentiation: dQ/dx ≈ [Q(1.701) − Q(1.699)] / 0.002 = −3.935
δQ = |−3.935| × 0.02 = 0.079
Sample mean: x̄ = (1/n) Σxᵢ
Standard error (uncertainty of the mean): δx̄ = s / √n
General error propagation: δQ = √[ Σᵢ (∂Q/∂xᵢ · δxᵢ)² ]
Product/quotient shortcut (relative uncertainties): δQ/Q = √[ (δa/a)² + (δb/b)² + ... ]
⚠️ The standard error (s/√n) and the standard deviation (s) are different quantities. The exam will almost certainly test whether you know which to use. The standard error is the uncertainty in the mean.
⚠️ When propagating uncertainty through trig functions, always convert angle uncertainties to radians. The derivative of sin(θ) is cos(θ) only when θ is in radians.
⚠️ For complex multi-variable functions, numerical partial derivatives (finite differences) are perfectly acceptable and often faster than working out the algebra by hand.
⚠️ The relative-uncertainty shortcut only works cleanly for pure products and quotients. If there are additions or subtractions in the formula, you must use the full partial-derivative method.
Q: What is the difference between standard deviation and standard error?
A: Standard deviation (s) measures the spread of individual measurements. Standard error (s/√n) measures the uncertainty in the sample mean. Standard error is always smaller by a factor of √n.
Q: You measure a quantity five times and get s = 3.0. What is the uncertainty of your best estimate?
A: δx̄ = 3.0 / √5 = 1.34.
Q: A volume is computed as V = L × W × H. The relative uncertainty in L is 0.32%, in W is 0.24%, and in H is 0.33%. What is the relative uncertainty in V?
A: δV/V = √(0.32² + 0.24² + 0.33²) % = √(0.1024 + 0.0576 + 0.1089) % = √0.2689 % = 0.52%.
Q: When propagating uncertainty through Q = f(x, y) where θ = 40° ± 3°, what value do you use for δθ?
A: Convert to radians: δθ = 3 × π/180 = 0.0524 rad.
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