Source: Final Exam Formula Booklet
Tags: error propagation, uncertainty, standard deviation, variance, standard error, SDOM, z-score, confidence interval, normal distribution, z-table, ENGR 216, PHYS 216, Texas A&M
This material covers how to quantify and propagate measurement uncertainty, summarise data with variance and standard deviation, and use the normal distribution to build confidence intervals for a population mean. The error propagation formula (partial derivatives method) and the confidence interval formula are the two heaviest computational pieces on the exam.
Uncertainty (δq)
The estimated range within which the true value of a measured or calculated quantity is expected to lie. Propagated through functions using partial derivatives.
Population variance (σ²)
The average of the squared deviations from the population mean μ. Divides by n (the full population size).
Sample variance (s²)
The average of the squared deviations from the sample mean x̄. Divides by (n − 1) to correct for bias (Bessel's correction).
Standard deviation (σ or s)
The square root of the variance. Same units as the original data.
Standard error of the mean (SDOM)
The standard deviation of the sampling distribution of the mean. Equals standard deviation / √n. Tells you how precisely you know the mean, not how spread out individual measurements are.
Z-score
The number of standard deviations a value sits away from the mean: z = (x − μ) / σ. Converts any normal distribution to the standard normal.
Confidence interval
A range of values, calculated from sample data, that is expected to contain the true population mean with a specified probability (confidence level).
Confidence level (1 − α)
The probability that the confidence interval contains the true mean. Common levels: 90% (α = 0.10), 95% (α = 0.05), 99% (α = 0.01).
If q = f(x, ..., z), then the uncertainty in q is:
δq = √[ (∂q/∂x · δx)² + ... + (∂q/∂z · δz)² ]
Steps to apply:
Write the function q in terms of the measured variables
Take the partial derivative of q with respect to each variable
Multiply each partial derivative by the uncertainty in that variable
Square each term, sum them, take the square root
This assumes the uncertainties are independent and random.
dy/dx = (dy/du)(du/dx)
Useful when q depends on a variable through an intermediate expression.
Population (you have every data point)
Variance: σ² = (1/n) Σ(xᵢ − μ)²
Standard deviation: σ = √σ²
Uses population mean μ and divides by n
Sample (you have a subset of the population)
Variance: s² = (1/(n−1)) Σ(xᵢ − x̄)²
Standard deviation: s = √s²
Uses sample mean x̄ and divides by (n − 1)
The (n − 1) denominator corrects the downward bias you get when estimating spread from a sample.
SDOM = standard deviation / √n
This measures the uncertainty in the sample mean, not in individual data points. As sample size grows, SDOM shrinks.
X ~ N(μ, σ²) means X follows a normal distribution with mean μ and variance σ².
For a single observation:
z = (x − μ) / σ
For a sample mean:
z = (x̄ − μ) / (σ / √n)
The z-score transforms any normal distribution to the standard normal N(0, 1), allowing you to use the z-table.
The exam provides two versions of the z-table:
Cumulative (left-tail) table: gives P(Z ≤ z), the area under the curve to the left of z
Area-between-0-and-z table: gives P(0 ≤ Z ≤ z), the area between the centre and z
Know which table you are using before looking up a value. For a two-sided confidence interval, you need z_{α/2}, the z-value that leaves α/2 in each tail.
Common z_{α/2} values:
90% CI (α = 0.10): z_{0.05} = 1.645
95% CI (α = 0.05): z_{0.025} = 1.96
99% CI (α = 0.01): z_{0.005} = 2.576
For a sample of size n with known population standard deviation σ:
x̄ − z_{α/2} (σ/√n) ≤ μ ≤ x̄ + z_{α/2} (σ/√n)
This is a two-sided interval. The confidence level is 100(1 − α)%.
Width of the confidence interval:
Width = U − L = 2 · z_{α/2} · (σ/√n)
To make the interval narrower: increase n or accept a lower confidence level.
Formula | Expression |
|---|---|
Error propagation | δq = √[ Σ (∂q/∂xᵢ · δxᵢ)² ] |
Population variance | σ² = (1/n) Σ(xᵢ − μ)² |
Sample variance | s² = (1/(n−1)) Σ(xᵢ − x̄)² |
SDOM | σ/√n or s/√n |
Z-score (single value) | z = (x − μ)/σ |
Z-score (sample mean) | z = (x̄ − μ)/(σ/√n) |
Confidence interval | x̄ ± z_{α/2} · σ/√n |
⚠️ Population statistics divide by n. Sample statistics divide by (n − 1). Mixing these up is one of the most common errors.
⚠️ The error propagation formula requires partial derivatives. If you are shaky on partial differentiation, practise before the exam.
⚠️ SDOM ≠ standard deviation. SDOM = SD/√n. It tells you the precision of the mean, not the spread of the data.
⚠️ When using the z-table, check whether it gives the left-tail area or the area between 0 and z. The exam provides both, and picking the wrong one shifts your answer.
⚠️ The confidence interval formula on this exam assumes σ is known. If the problem gives you s (sample standard deviation) and tells you to treat it as σ, use it directly in the formula.
⚠️ The exam instructions say: do not round intermediate results. Keep full precision until the final answer, then round to the specified decimal places.
Q: You measure length L = 5.0 ± 0.2 cm and width W = 3.0 ± 0.1 cm. Area A = LW. What is δA?
A: ∂A/∂L = W = 3.0. ∂A/∂W = L = 5.0. δA = √[(3.0 × 0.2)² + (5.0 × 0.1)²] = √[0.36 + 0.25] = √0.61 = 0.781 cm².
Q: What is the difference between σ and s?
A: σ is the population standard deviation (divides by n, uses μ). s is the sample standard deviation (divides by n − 1, uses x̄).
Q: A sample of 25 measurements has x̄ = 10.3 and the population σ = 2.0. Find the 95% confidence interval for μ.
A: z_{0.025} = 1.96. Margin = 1.96 × 2.0/√25 = 1.96 × 0.4 = 0.784. CI: 10.3 − 0.784 ≤ μ ≤ 10.3 + 0.784, so 9.516 ≤ μ ≤ 11.084.
Q: What is the width of the confidence interval above?
A: Width = 2 × 0.784 = 1.568.
Q: If you quadruple the sample size from 25 to 100, what happens to the width of the CI?
A: The width is proportional to 1/√n. √100/√25 = 10/5 = 2. The width halves.
error propagation, uncertainty propagation, partial derivatives, quadrature, population variance, sample variance, standard deviation, Bessel's correction, standard error of the mean, SDOM, SEM, z-score, standard score, normal distribution, Gaussian, z-table, confidence interval, confidence level, alpha, margin of error, ENGR 216, PHYS 216, Texas A&M, final exam review