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Showing 1–30 of 1709 reference entries

Mathematics & logicExamples and counterexamples

Learn what a claim covers, explain why it works, and recognise an allowed case that shows it needs changing.

MethodMathematics & logicCounting and zero

Count each object once, explain the final count, and distinguish quantity, position and identifying labels. Make zero’s role explicit.

DefinitionMathematics & logicNumber lines and order

Use number-line positions to compare whole numbers. Read the scale, recognise equality, and distinguish numerical distance from drawing size.

DefinitionMathematics & logicAddition and subtraction

Combine separate counts, remove a part, and find differences or missing parts. Connect objects, equations and number-line moves, then use tens and ones to regroup without changing value.

DefinitionMathematics & logicMultiplication and division

Count equal groups, distinguish sharing from making groups, and check with multiplication. Split calculations into parts; interpret zero and remainders in context.

DefinitionMathematics & logicFractions

Choose a whole, count equal fractional units, and connect sharing with division. Place fractions on the number line, including values beyond one, and explain what their numerator and denominator mean.

DefinitionMathematics & logic · Physics · MeasurementRatios and rates

Distinguish part-to-part ratios, fractions of a whole and rates with units. Scale quantities together, compare unit rates, and explain when a relationship is proportional. Check zero cases, starting amounts and whether a result fits its context.

DefinitionMathematics & logic · Physics · MeasurementQuantities and units

Describe a quantity as a numerical value with a unit, and distinguish its physical kind from its dimension. Build length, area, volume and rate units, check meaningful combinations, and separate exact unit relations from measurement uncertainty.

DefinitionMathematics & logic · Physics · MeasurementUnit conversions

Convert quantities with exact unit ratios, track cancellation through compound units, and apply the correct powers to area and volume. Distinguish temperature readings from differences, check dimensions and context, and retain measurement uncertainty when reporting.

MethodMathematics & logic · Physics · MeasurementEstimation and uncertainty

Distinguish exact model values, rounded approximations and estimates from measurements. Declare the rounding rule, preserve information through calculations and unit changes, and interpret resolution, precision, accuracy and the stated uncertainty type without turning a report’s digits into guaranteed bounds.

DefinitionMathematics & logicVariables and expressions

Use symbols to name numbers or quantities, describe their allowed values, and evaluate expressions by replacing every occurrence consistently. Preserve grouping, signs, fractions and units, and distinguish a calculated value from solving an equation or proving two rules equivalent.

DefinitionMathematics & logicEquality and equations

Read equality as a relationship, find the permitted values that make an equation true, and justify one-step solutions by reversible operations on both sides. Work exactly with negative numbers and fractions, recognize all-solution and no-solution cases, and check the original equation, units and context.

DefinitionMathematics & logic · PhysicsCoordinates and tables

Locate a point from an ordered pair, read named axes and their scales, and connect the same values in words, tables and plots. Keep signs, fractions and units exact, distinguish an axis point from a quadrant point, and separate selected observations from a stated rule and its domain before drawing connections.

DefinitionMathematics & logic · PhysicsSlope and straight lines

Read a straight line as a stated relationship, calculate its slope from consistent signed coordinate changes, and interpret its intersections with the axes. Keep horizontal, vertical and coincident-point cases distinct, attach units and domain conditions to context models, and use numerical scales rather than appearance alone to judge a line’s change.

DefinitionMathematics & logic · Physics · MeasurementReading graphs

Read a graph’s quantities, units, ticks and context before judging its visual impression. Calculate differences, reference-based percentages and interval rates from the represented values; distinguish legitimate zoom from distorted spacing; and keep samples, interpolation, missing information, uncertainty and causal explanations separate.

MethodPhysics · MeasurementObservables and measurement

Choose the physical quantity you want to investigate, specify the point, event or location involved, and explain how a reading relates to that quantity.

MethodPhysicsSystems and boundaries

Choose what belongs to an investigation, describe its condition at a stated time, and identify interactions with what lies outside its boundary.

MethodPhysicsModels and assumptions

Choose what a model represents, make its simplifications explicit, and check whether its predictions answer the physical question you actually asked.

MethodPhysicsHypotheses, laws and theories

Identify what a scientific statement does, examine its evidence and conditions, and avoid treating its name as a score for certainty.

MethodMathematics & logic · PhysicsApproximations

Build a useful estimate, compare a correction with a stated reference scale, and check what is lost when you leave out a term.

ApproximationMathematics & logic · PhysicsSymmetry and invariance

Specify a transformation, identify what stays unchanged, and distinguish symmetry of an object from invariance of a physical model.

Mathematical statementPhysicsConservation laws

Use a defined system and a named quantity to track transfers, distinguish changing parts from conserved totals, and check the conditions of energy and momentum balances.

ModelPhysicsEmergence and levels of description

Choose a descriptive level, connect collective quantities with component behavior, and check when a simpler physical description answers a specified question.

ModelMathematics & logic · PhysicsVectors

Describe displacement with fixed-axis components, combine compatible vectors, and use small exact examples to introduce span, independence, basis and real vector spaces.

DefinitionPhysics · Information · MeasurementScientific records

Build an inspectable physics record, test and revise a bounded cart model, preserve processing history, and plan findable data with appropriate sharing conditions.

MethodPhysicsPosition, velocity and acceleration

Describe a point's motion on a chosen straight axis, then connect signed changes, instantaneous rates and one exact set of position, velocity and acceleration graphs.

DefinitionPhysicsNewton’s laws and forces

Choose one body, identify the forces acting on it, and use their vector sum to distinguish acceleration, balanced motion and mutual interactions.

ModelPhysicsWork, energy and power

Connect force and displacement to changes in motion, follow a spring’s stored energy, and distinguish an energy transfer from its rate.

ModelPhysics · MeasurementSpeed of light in vacuum

c denotes the speed of light in vacuum. Its numerical value is fixed exactly in SI units.

DefinitionPhysics · MeasurementPlanck constant

h denotes the Planck constant. Its numerical value is fixed exactly in SI units.

Definition
ModelSource checked

Linear regression coefficients and covariance

Mathematics & logic

For fixed full-column-rank X and Y=Xβ+ε with Eε=0 and Cov ε=σ²I, least squares gives β̂=(XᵀX)⁻¹XᵀY, Eβ̂=β and Cov β̂=σ²(XᵀX)⁻¹.

Key knowledge

For fixed full-column-rank X and Y=Xβ+ε with Eε=0 and Cov ε=σ²I, least squares gives β̂=(XᵀX)⁻¹XᵀY, Eβ̂=β and Cov β̂=σ²(XᵀX)⁻¹. Independent normal errors additionally make β̂ exactly multivariate normal.

Conditions and scope

  • The mean relationship and error covariance are model assumptions.
  • Predictor measurement error, dependence or unequal error variance require a different uncertainty model; numerical QR avoids explicit normal-equation inversion.
Sources · 1

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. STATS 200, Lecture 25: The linear modelZhou Fan / Stanford UniversityCourse notes · PDFweb.stanford.eduRead source

Evidence and review

Lecture 25 fixed-design model, likelihood/least-squares formula and exact Gaussian distribution footnote.

2026-10-10 · Independent AI editorial review: All241 compact claims, hypotheses, displayed equations/conventions and all596 origin semantic links read; entire241-entry author exact-source inspection matrix and declared limitations reviewed. Prior105 actual inherited Mathematics/Information records read for whole-inventory breadth. Targeted independent primary sections are listed below; this does NOT claim every cited source body or every proof was independently re-read. This is AI reference-content review, not human disciplinary certification, exhaustive textbook reproduction or experimental replication. Receipt: mathematics-information-independent-content-review.1.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–12 of 12 relationships

Scientific relationships · 12 / 12

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • For fixed full-column-rank X and Y=Xβ+ε with Eε=0 and Cov ε=σ²I, least squares gives β̂=(XᵀX)⁻¹XᵀY, Eβ̂=β and Cov β̂=σ²(XᵀX)⁻¹. Independent normal errors additionally make β̂ exactly multivariate normal.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • For fixed full-column-rank X and Y=Xβ+ε with Eε=0 and Cov ε=σ²I, least squares gives β̂=(XᵀX)⁻¹XᵀY, Eβ̂=β and Cov β̂=σ²(XᵀX)⁻¹. Independent normal errors additionally make β̂ exactly multivariate normal.
  3. Uses definition
    Relationship sources · 1

    Linked statements

    • For fixed full-column-rank X and Y=Xβ+ε with Eε=0 and Cov ε=σ²I, least squares gives β̂=(XᵀX)⁻¹XᵀY, Eβ̂=β and Cov β̂=σ²(XᵀX)⁻¹. Independent normal errors additionally make β̂ exactly multivariate normal.
  4. Uses definition
    Relationship sources · 1

    Linked statements

    • Residuals eᵢ=yᵢ−f(xᵢ;β̂) compare observed and fitted responses. Patterns against predictors, fitted values or run order can expose functional misspecification, changing variance or dependence. A high R² does not certify model adequacy.
  5. Uses definition
    Relationship sources · 2

    Linked statements

    • For k groups let B=Σnᵢ(x̄ᵢ−x̄)² and W=Σᵢⱼ(xᵢⱼ−x̄ᵢ)². Under equal means, independent normal observations with common variance give F=[B/(k−1)]/[W/(n−k)] the F_(k−1,n−k) law; reject for large F.
  6. Related to
    Relationship sources · 2

    Linked statements

    • An instrument calibration curve models indication y as a function of reference input x, such as y=a+bx+ε. Conventional ordinary-least-squares calibration inference assumes adequately negligible reference-input uncertainty, independent response errors and a suitable constant-variance model. If these fail, fitting and uncertainty assessment must be adapted to the actual error model, for example weighted or errors-in-variables methods where appropriate.

    Operational indication-on-reference fitting requires the stated regression error and input-uncertainty assumptions.

  7. Related to
    Relationship sources · 1

    Linked statements

    • For identical sample aliquots diluted to a common final volume, blank-corrected linear response satisfies S=k(c₀V₀+cstdVstd)/Vf. Regression against added concentration can infer the original sample concentration while calibrating in its matrix.

    A fitted line estimates slope and intercept.

  8. Related to
    Relationship sources · 1

    Linked statements

    • For parent P decaying into daughter D at constant λ, normalization to a stable isotope S gives D/S=(D/S)₀+(P/S)(e^(λt)−1). Coeval samples with a common initial daughter ratio, differing parent ratios and closed subsequent evolution define an isochron with slope m=e^(λt)−1 and age t=ln(1+m)/λ. A fitted straight line by itself does not prove those geological assumptions.

    A fitted line requires a suitable error model; shared denominator covariance matters.

  9. Related to
    Relationship sources · 2

    Linked statements

    • Ocean-colour band-ratio models estimate near-surface chlorophyll-a from water-leaving remote-sensing reflectances. In the OC family, X=log₁₀(Rrs_blue/Rrs_green) uses a specified maximum blue band and a green band, and log₁₀(CHL/(1mg m⁻³))=Σ_(j=0)^4a_jX^j. Coefficients and bands are empirically calibrated for a sensor/model version; this is a retrieval relationship rather than a direct count of phytoplankton.

    An empirical coefficient model needs applicable statistical validation.

  10. Related to
    Relationship sources · 1

    Linked statements

    • For predictors h, population risk is expected loss under a declared data distribution; empirical risk averages loss over a sample. Supervised learning observes targets, while unsupervised objectives describe structure or latent representations without guaranteeing a uniquely meaningful interpretation. Minimizing sample loss is a statistical estimation problem as well as an optimization problem; target population and sampling assumptions determine what generalization means.

    Least-squares regression supplies one supervised empirical loss.

  11. Related to
    Relationship sources · 2

    Linked statements

    • Connected order-parameter correlations define a correlation length ξ. Near a continuous critical point, ξ∝|t|⁻ν and the critical correlator scales as r^(−d+2−η) within its scaling window, t=(T−T_c)/T_c. Gaussian theory gives ν=1/2 and η=0; fluctuation corrections depend on dimension, symmetry and interaction range.

    Exponent inference requires a justified fit interval and uncertainty.

  12. Related to
    Relationship sources · 1

    Linked statements

    • A calibration likelihood can treat reference values x_i as latent uncertain quantities, with measured standards r_i and instrument indications y_i modelled jointly around x_i and f(x_i,θ). Posterior inference p(θ,x|r,y) combines the joint likelihood with documented priors; factorizing reference and instrument likelihoods additionally requires conditional independence. Ignoring reference uncertainty can bias slopes and understate inverse-calibration uncertainty.

    Ordinary least squares is only a restricted likelihood case.

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math-linear-regression
Revision
university-core.1
Structured reference record
University core expansion