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Astronomical Distance Units .

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Balancing Chemical Equations

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Integration By Parts

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Introduction to Conformal Field Theory .

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Similar Matrices and Diagonalization .

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Spherical Trigonometry

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Symmetric Groups

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Taylor and Maclaurin Series .

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The Integers Modulo n Under + and x

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Vector Calculus

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Volume Integrals

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Field Dimensions

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Spin 1 Eigenvectors .

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The Vacuum Catastrophe

Probability and Statistics

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Chebyshev's Theorem

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Factor Analysis

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Hypothesis Testing

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Linear Regression

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Monte Carlo Methods

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Pearson Correlation

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Permutations and Combinations

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Pooled Variance and Standard Error

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Probability Rules

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Sample Size Determination

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Sampling Distributions

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Lagrangians in Quantum Field Theory

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Path Integral Formulation

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Relativistic Quantum Field Theory

Quantum Mechanics

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Basic Relationships

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Clebsch-Gordan Coefficients .

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Electron Orbital Angular Momentum and Spin

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Entangled States

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Heisenberg Uncertainty Principle

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Ladder Operators .

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Multi Electron Wavefunctions

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Pauli Exclusion Principle

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Pauli Spin Matrices

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Photoelectric Effect

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Position and Momentum States

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Probability Current

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Schrodinger Equation for Hydrogen Atom

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Spin 1/2 Eigenvectors

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The Differential Operator

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Time Dependent Perturbation Theory

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Time Evolution and Symmetry Operations

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Time Independent Perturbation Theory

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Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids .

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Fermi-Dirac Statistics .

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Intrinsic and Extrinsic Semiconductors

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The P-N Junction

Special Relativity

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Electromagnetic 4 - Potential

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Energy and Momentum, E = mc²

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Lorentz Invariance

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Lorentz Transform

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Lorentz Transformation of the EM Field

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Newton versus Einstein

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Spinors - Part 1 .

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Spinors - Part 2 .

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The Lorentz Group

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Velocity Addition

Statistical Mechanics

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Black Body Radiation

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Entropy and the Partition Function

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The Harmonic Oscillator

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String Theory

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Bosonic Strings

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Extra Dimensions

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Introduction to String Theory

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Kaluza-Klein Compactification of Closed Strings

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Strings in Curved Spacetime

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Toroidal Compactification

Superconductivity

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Introduction to Superconductors

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Superconductivity (Lectures 1 - 10)

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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields

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Generators of a Supergroup

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Grassmann Numbers

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Introduction to Supersymmetry

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The Gauge Hierarchy Problem

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)

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Gauge Theories (Yang-Mills)

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Gravitational Force and the Planck Scale

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Introduction to the Standard Model

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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge

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Quantum Flavordynamics and Quantum Chromodynamics

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Special Unitary Groups and the Standard Model - Part 1 .

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Special Unitary Groups and the Standard Model - Part 2

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Special Unitary Groups and the Standard Model - Part 3 .

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Standard Model Lagrangian

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The Higgs Mechanism

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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Last modified: January 26, 2018


Pooled Variance and Standard Error
----------------------------------

The pooled variance and standard error are used in tests of 
significance for differences in means or proportions between 
2 independent populations.

Equal Variances
---------------

If we assume that the sample variance are the same we can use 
the poooled estimate of the variance by using the following 
formula:

s_p² = [(n₁ - 1)s₁² + (n₂ - 1)s₂²]/[(n₁ - 1) + (n₂ - 1)]

df = (n₁ - 1) + (n₂ - 1)

   = n₁ + n₂ - 2

This is just the weighted average of s₁² + s₂².  When n₁ = n₂ 
we get s_p² = (s₁² + s₂²)/2.

Under the assumption of equal population variances, the pooled 
sample variance provides a higher precision estimate of variance 
than the individual sample variances. This higher precision can 
lead to increased statistical power when used in statistical 
tests that compare the populations, such as the t-test.

The F test can be used to determine whether the sample variances 
are different or not.

Pooled Standard Error
---------------------

The pooled standard error for equal variances is:

SE = s_p√(1/n₁ + 1/n₂)

Unequal Variances
-----------------

When the sample variances are different we can’t get a pooled 
estimate.  Instead, we use a standard error formula that includes 
both standard deviations, but separately.  We use:

SE = √(s₁²/n₁ + s₂²/n₂)

SE₁ = s₁/√n₁ = √(s₁²/n₁) and SE₂ = s₂/√n₂ = √(s₂²/n₂)

In this case the df is rather complicated.  It is the integer
part of:

              √(s₁²/n₁ + s₂²/n₂)
df = ---------------------------------------
     ((s₁²/n₁)²/(n₁ - 1) + ((s₂²/n₂)²/(n₂ - 1)

This is the Satterthwaite correction.  One can always use this
regardless of whether the variances are equal or not.

In the situation where we are considering proportions we make
the following substitutions:

s² = pq = p(1 - p)

s = √(pq)

For a single proportion the SE is:

SE =  √(pq)/√n = √(pq/n)