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Quaternions 2 .

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Spin 1 Eigenvectors .

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Factor Analysis

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Hypothesis Testing

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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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Electron Orbital Angular Momentum and Spin

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Multi Electron Wavefunctions .

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Spin 1/2 Eigenvectors

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Solid State Electronics

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Band Theory of Solids .

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The P-N Junction

Special Relativity

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Electromagnetic (Faraday) Tensor .

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Energy and Momentum in Special Relativity, E = mc² .

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Lorentz Invariance .

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Lorentz Transform .

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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 Continuity Equation .

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Statistical Mechanics

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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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Strings in Curved Spacetime

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Toroidal Compactification

Superconductivity

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Bardeen–Cooper–Schrieffer Theory

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Introduction to Superconductivity .

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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)