Experimental Physics & Data Analysis I — formula sheet
Every boxed result from the 17 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
Expectation, Variance and Their Algebra
The mean of a linear combination is the same linear combination of the means, always — dependence is irrelevant. The variance is the second moment minus the square of the first. Adding a constant slides the whole distribution and leaves the spread untouched; multiplying by a stretches deviations by |a|, so variance (a squared quantity) scales by a².
Variance of Linear Combinations and Covariance
The spread of a weighted sum is a weighted sum of all pairwise spreads — each variable with itself (variance) and each pair with the other (covariance). Weights enter squared on the diagonal, so scaling a variable by a scales its variance contribution by a2. Off-diagonal covariance terms can be positive (uncertainties reinforce) or negative (they partially cancel). When inputs are uncorrelated, only the diagonal survives and variances simply add.
First-Order Propagation of Uncertainty
Each input contributes a term equal to its own variance amplified by the squared sensitivity ∂f/∂xi — how strongly f responds to a nudge in xi. Because the terms are squares they always add (uncertainties never cancel), which is why independent errors combine "in quadrature": the total is the root-sum-of-squares, dominated by the largest single contribution.
Propagation with the Covariance Matrix
The output covariance is a two-sided congruence of the input covariance by the Jacobian: J maps the “shape” of the input uncertainty ellipsoid into output space, and the transpose on the right guarantees the result is again a symmetric, positive-semidefinite matrix. Diagonal entries of Σy are the variances of each yi; off-diagonal entries are the induced covariances — even if the inputs were uncorrelated, sharing input variables through J makes the outputs correlated.
The Standard Error of the Mean
Averaging N independent measurements does not change the typical size of a single error, σ, but it reduces the error on the average by a factor √N. The gain has diminishing returns: going from 1 to 4 measurements halves the error, but going from 100 to 400 also only halves it. The mean converges to the true value μ, but slowly.
Bessel's Correction for Sample Variance
Averaged over all possible samples of size N, the Bessel-corrected sample variance equals the true population variance exactly — no bias, for any parent distribution with finite variance. Dividing instead by N would give E = σ2(N−1)/N < σ2, an underestimate. The single missing unit in N−1 is the degree of freedom spent locating X̄.
The Inverse-Variance Weighted Mean
Weight each measurement by its precision, the reciprocal of its variance, then renormalise. The combined estimate's precision is the sum of the individual precisions — informations add, so the combined uncertainty is always smaller than that of the best single input.
The Binomial Distribution from Bernoulli Trials
The chance of exactly k successes is the probability of one favourable arrangement, pkqN−k, multiplied by the number of favourable arrangements, NCk. The count drifts around Np; its spread, measured by the standard deviation √[Np(1−p)], is largest at p=½ and pinches to zero as p approaches 0 or 1.
Poisson Statistics of Counting
The probability of observing exactly k counts when the expected number is μ is μke−μ/k!. The distribution is fixed by the single parameter μ; both its centre and its spread grow with μ, but the spread grows only as the square root, so larger counts are proportionally sharper. A raw count N is its own estimate of μ, with uncertainty √N.
Characteristic Functions and Moments
The characteristic function packages the whole law of X into one bounded function of a real dial t. Turning the dial slowly near zero and reading off successive slopes recovers the moments one at a time: the first derivative gives i times the mean, the second gives −1 times the mean square, and so on. And because it is a Fourier transform, adding independent randomness — which convolves densities — simply multiplies the transforms.
The Central Limit Theorem
Add up many independent finite-variance contributions, subtract the mean, and rescale by σ√N; whatever the individual variables looked like, the result is a standard bell curve. All the parent distribution’s idiosyncratic features — skewness, kurtosis, discreteness — are relegated to higher-order corrections that die as powers of 1/√N. Only the first two moments, μ and σ2, survive; everything else is forgotten.
The Method of Maximum Likelihood
The maximum-likelihood estimate is the parameter value at which the total score — the summed sensitivity of each data point's log-density to θ — cancels to zero. Each observation "pulls" θ in the direction that would make that observation more probable; the MLE is the balance point where all the pulls sum to nothing. A negative second derivative confirms it is a peak, and its magnitude sets the estimator's precision.
Least Squares from Gaussian Likelihood
Fitting by weighted least squares is not a convenient approximation to the "real" statistical answer — under independent Gaussian errors it is the maximum-likelihood answer, exactly. Each residual is weighted by 1/σi², so precise points (small σ) pull the fit hard and noisy points count for little.
Linear Least Squares and the Normal Equations
The best-fit parameters are a linear map applied to the data. The matrix ATWA (the "curvature" or Fisher information matrix) measures how sharply χ2 rises away from the minimum; its inverse is the parameter covariance, so directions the data constrain poorly show up as large eigenvalues of (ATWA)−1. The right-hand side ATWy is the weighted projection of the data onto each basis function.
Parameter Uncertainties and the Cramer-Rao Bound
The Fisher matrix is the average sharpness of the likelihood peak — large curvature (a narrow, steep peak) means the data pin the parameters down tightly, so the inverse (the covariance) is small. Off-diagonal elements encode parameter degeneracy: a shallow direction of the likelihood surface (near-singular I) becomes a long axis of the error ellipse. No unbiased estimator can do better than I−1, and maximum likelihood reaches this floor for large samples. The diagonal σj = √(I−1)jj is the marginal 1σ error on θj (with the others floating); 1/√Ijj is the tighter error with the others held fixed.
The Chi-Squared Distribution
A sum of k independent squared standard-normals is right-skewed, supported on the positive axis, and peaks near x = k − 2. Its centre of mass sits at k and its spread grows as √(2k); each extra degree of freedom adds one unit of expected value and two units of variance. The xk/2−1 factor controls the behaviour near the origin and the e−x/2 factor the exponential tail.
Goodness of Fit and Degrees of Freedom
A well-fitting model over N points with M free parameters should return χ²min near N − M, i.e. χ²ν ≈ 1 ± √(2/ν). Substantially larger means the model is wrong or the errors are underestimated; substantially smaller means the errors are overestimated or the model is overfitting. Each parameter you fit costs exactly one degree of freedom because fitting projects the residual out of one more dimension.