Tags: statistics, uncertainty propagation, z-score, confidence interval, standard error, sample mean, sample size, normal distribution, error analysis, ENGR 216, Texas A&M
This covers the statistical foundations tested in ENGR 216: computing sample means and standard errors, using the normal distribution and z-scores to find probabilities, constructing confidence intervals, determining required sample sizes, and propagating measurement uncertainty through formulas. These topics appear across multiple sections of the final and account for a large share of the marks.
Sample mean (x-bar)
The arithmetic average of a set of n measurements. Calculated as the sum of all values divided by n.
Sample standard deviation (s)
A measure of spread in a data set. Quantifies how much individual measurements deviate from the sample mean. Uses n - 1 in the denominator (Bessel's correction).
Standard error of the mean (SEM)
The uncertainty of the best estimate of the mean. Calculated as s / sqrt(n). Tells you how precisely you know the population mean from your sample.
Z-score
The number of standard deviations a data point sits from the mean: z = (x - mu) / sigma. Positive z means above the mean, negative means below.
Confidence interval (CI)
A range of values, constructed from sample data, that is expected to contain the true population mean at a given confidence level. For a known population standard deviation: x-bar +/- z * (sigma / sqrt(n)).
Margin of error (E)
The half-width of a confidence interval. E = z * (sigma / sqrt(n)).
Normal distribution
A symmetric, bell-shaped probability distribution fully described by its mean (mu) and standard deviation (sigma). Many physical measurements follow this distribution.
Uncertainty propagation (RSS method)
For a quantity Q that depends on measured variables, the combined uncertainty is the root sum of squares of each partial derivative multiplied by its variable's uncertainty. Assumes independent, random errors.
For n data values, x-bar = (x1 + x2 + ... + xn) / n
Works with symbolic expressions too. If values are written in terms of a parameter (e.g. a + 3, a, a + 4, a + 5, a + 2), sum the constants separately from the parameter terms, then divide by n
Example: five values summing to 5a + 14 give x-bar = a + 2.8
Formula: SEM = s / sqrt(n)
Example: s = 2.41, n = 10, so SEM = 2.41 / sqrt(10) = 0.76
The divisor is always the square root of the sample size. For n = 10, you divide by sqrt(10)
A larger sample size reduces the standard error (better precision on your estimate of the mean)
z = (x - mu) / sigma
Example: parachute opening altitude with mu = 200 m, sigma = 30 m. For x = 100 m: z = (100 - 200) / 30 = -3.33
Example: resistor with mu = 10 ohms, sigma = 0.2 ohms. For x = 10.256 ohms: z = (10.256 - 10) / 0.2 = 1.28
To find the probability between two values:
Calculate z-scores for both boundaries
Look up each cumulative probability in the standard normal table
Subtract the smaller from the larger
Example: force with mu = 15.0 kips, sigma = 1.25 kips. P(10 < x < 12): z1 = (10 - 15)/1.25 = -4.0, z2 = (12 - 15)/1.25 = -2.4. P(z < -2.4) = 0.0082, P(z < -4.0) is approximately 0. Result: 0.0082
Lower bound: x-bar - z * (sigma / sqrt(n))
Upper bound: x-bar + z * (sigma / sqrt(n))
Example: n = 36, x-bar = 2.6 g/mL, sigma = 0.3 g/mL, z = 1.96 for 95% confidence. Lower bound = 2.6 - 1.96 * (0.3 / sqrt(36)) = 2.6 - 0.098 = 2.50
Effect of changing confidence level:
Decreasing the confidence level (e.g. 95% to 90%) decreases the z-critical value
This reduces the margin of error, making the interval narrower
The interval does not shift; only the width changes
Formula: n = (z * sigma / E)^2, always round up to the next whole number
Example: z = 1.96, sigma = 0.3, E = 0.05. n = (1.96 * 0.3 / 0.05)^2 = (11.76)^2 = 138.30, rounded up to 139
For a function Q(x, y, theta, ...):
delta-Q = sqrt[ (dQ/dx delta-x)^2 + (dQ/dy delta-y)^2 + (dQ/dtheta * delta-theta)^2 + ... ]
Key steps:
Compute the nominal value of Q
Find the partial derivative of Q with respect to each measured variable
If a variable is in degrees, convert its uncertainty to radians: delta-theta(rad) = delta-theta(deg) * pi / 180
Multiply each partial derivative by its variable's uncertainty
Take the root sum of squares
Example (volume of a sphere, V = 4/3 pi r^3):
dV/dr = 4 pi r^2
This is the partial derivative needed for propagation
Example (Q = (x + 2)(y + 1) sin(theta)):
dQ/dx = (y + 1) sin(theta)
dQ/dy = (x + 2) sin(theta)
dQ/dtheta = (x + 2)(y + 1) cos(theta)
With x = 10 +/- 2, y = 7 +/- 1, theta = 40 deg +/- 3 deg:
Nominal Q = 12 8 sin(40) = 61.71
delta-theta = 3 * pi/180 = 0.05236 rad
delta-Q = sqrt[(5.142 2)^2 + (7.714 1)^2 + (73.54 * 0.05236)^2] = sqrt[105.76 + 59.51 + 14.82] = 13.42
The uncertainty propagation formula assumes all measurement errors are independent and random. This is a true/false point that comes up on the exam.
Sample mean: x-bar = (1/n) * sum(xi)
Standard error: SEM = s / sqrt(n)
Z-score: z = (x - mu) / sigma
Confidence interval: x-bar +/- z * (sigma / sqrt(n))
Sample size: n = (z * sigma / E)^2 (round up)
Uncertainty propagation: delta-Q = sqrt[ sum( (dQ/dxi * delta-xi)^2 ) ]
Volume of a sphere: V = 4/3 pi r^3, so dV/dr = 4 pi r^2
⚠️ For sample size calculations, always round up to the next integer, never round down or to the nearest.
⚠️ When propagating uncertainty for angles, you must convert the angle uncertainty to radians before plugging into the formula.
⚠️ The standard error divisor is sqrt(n), not n. The exam tests this directly in fill-in-the-blank form.
⚠️ Decreasing confidence level makes the interval narrower, not wider. This is a common trap in multiple choice.
⚠️ The RSS uncertainty formula requires independent, random errors. This is tested as a true/false question.
⚠️ Know the partial derivative of common functions: the sphere volume derivative dV/dr = 4pir^2 is tested directly.
Q: A set of 10 measurements has a sample standard deviation of 2.41. What is the standard error of the mean?
A: SEM = 2.41 / sqrt(10) = 0.76.
Q: A normal distribution has mean 200 m and standard deviation 30 m. What is the z-score for a value of 100 m?
A: z = (100 - 200) / 30 = -3.33.
Q: If a 95% confidence interval is recalculated at 90% confidence with the same sample, what happens to its width?
A: It becomes narrower, because the z-critical value decreases.
Q: You want to estimate a mean within 0.05 at 95% confidence, with sigma = 0.3. What sample size do you need?
A: n = (1.96 * 0.3 / 0.05)^2 = 138.30, rounded up to 139.
Q: What is the partial derivative of V = 4/3 pi r^3 with respect to r?
A: dV/dr = 4 pi r^2.
Q: True or false: the uncertainty propagation formula assumes all measurement errors are independent and random.
A: True.
sample mean, arithmetic mean, x-bar, standard deviation, standard error of the mean, SEM, uncertainty of the best estimate, z-score, standard score, normal distribution, Gaussian distribution, bell curve, confidence interval, CI, margin of error, sample size determination, uncertainty propagation, error propagation, root sum of squares, RSS, partial derivatives, ENGR 216, experimental physics lab, Texas A&M