Lecture 22: Expectation, Variance, and Error

The Hook

You measure the C-C bond length in ethane. Ten measurements:

1.532, 1.528, 1.535, 1.530, 1.527, 1.534, 1.529, 1.531, 1.533, 1.528 A

Questions: What is THE bond length? How confident are you?

The uncomfortable answer: There is no single "true" value you can access. Every measurement has uncertainty. The best you can do is characterize the distribution of possible values.

The Fundamental Distinction

-13.6 eV

Calculated (exact in model)

-13.598 +/- 0.002

Observed (uncertain)

Source	Type	Reducible?
Instrument precision	Systematic/Random	Partially
Environmental fluctuation	Random	Partially
Sampling variation	Random	With more data
Quantum uncertainty	Fundamental	No
Model approximation	Systematic	With better theory

Sample Mean vs Population Mean

The sample mean x estimates the true mean mu. With more samples, our estimate becomes more precise.

Sampling from a Normal Distribution

Sample Mean x

Sample Std s

Std Error SE

95% CI contains mu?

True mean mu: 5

True std sigma: 1

Sample size n:

Standard Error = s/sqrt(n)

More samples reduce SE, making the mean estimate more precise. The 95% CI should contain mu about 95% of the time.

Expectation and Variance

Expectation (Mean):

$$\langle X \rangle = \sum_i x_i P(x_i) \quad \text{or} \quad \int x \, f(x) \, dx$$

Variance:

$$\sigma^2 = \langle (X - \mu)^2 \rangle = \langle X^2 \rangle - \langle X \rangle^2$$

Example: Fair Die

X = number rolled

E[X] = (1+2+3+4+5+6)/6 = 3.5 (but you can never roll 3.5!)
E[X^2] = (1+4+9+16+25+36)/6 = 15.17
Var(X) = 15.17 - 12.25 = 2.92
sigma = 1.71

Error Propagation

General Formula (independent uncertainties):

$$\delta f = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2 \delta x^2 + \left(\frac{\partial f}{\partial y}\right)^2 \delta y^2 + \cdots}$$

Error Propagation Calculator

Operation:

x =

+/- delta x =

y =

+/- delta y =

n (for power) =

Common Cases

Operation	Formula
f = x +/- y	delta f = sqrt(delta x^2 + delta y^2)
f = x * y or x/y	delta f/f = sqrt((delta x/x)^2 + (delta y/y)^2)
f = x^n	delta f/f = \|n\| * delta x/x
f = e^x	delta f = \|f\| * delta x
f = ln(x)	delta f = delta x / x

Adding uncertainties linearly is WRONG!

delta(x+y) != delta x + delta y

delta(x+y) = sqrt(delta x^2 + delta y^2) (add in quadrature)

Monte Carlo Error Propagation

When formulas get complicated, simulate! Generate random samples and see the distribution of f.

Monte Carlo: f = x^2/y with x = 4.0 +/- 0.2, y = 2.0 +/- 0.1

MC Mean

MC Std

Analytical f

Analytical delta f

Number of MC samples:

Monte Carlo naturally handles non-linear functions, correlated uncertainties, and non-Gaussian distributions.

Weighted Averages

When measurements have different precisions, weight by 1/sigma^2.

$$\bar{x}_w = \frac{\sum_i w_i x_i}{\sum_i w_i} \quad \text{where } w_i = \frac{1}{\sigma_i^2}$$

Weighted Average Calculator

Weighted Avg

Uncertainty

Simple Avg

The more precise measurement dominates the weighted average!

Summary

Concept	Symbol	Meaning
Expectation	E[X] or mu	Probability-weighted average
Variance	sigma^2	E[(X - mu)^2]
Standard deviation	sigma	sqrt(Var)
Standard error	SE	sigma/sqrt(n)

Every measurement is uncertain. Reporting a value without uncertainty is incomplete. Understanding error propagation connects calculated predictions to experimental reality.