[ { "name": "Quadratic formula", "subject": "math", "topic": "polynomial equations", "concepts": ["quadratic equation", "roots", "discriminant"], "latex": "x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}", "variables": [ { "symbol": "x", "meaning": "root of the equation ax^2 + bx + c = 0", "si_unit": null }, { "symbol": "a", "meaning": "coefficient of the quadratic term, with a not equal to 0", "si_unit": null }, { "symbol": "b", "meaning": "coefficient of the linear term", "si_unit": null }, { "symbol": "c", "meaning": "constant term", "si_unit": null } ], "tags": ["algebra", "roots", "polynomial"], "description": "Gives both roots of the quadratic equation ax^2 + bx + c = 0 from its coefficients; the discriminant b^2 - 4ac decides whether the roots are real and distinct, repeated, or complex.", "source": "standard result; see e.g. Stewart, Redlin & Watson, Precalculus" }, { "name": "Distance formula in the plane", "subject": "math", "topic": "coordinate geometry", "concepts": ["Euclidean distance", "Pythagorean theorem", "Cartesian coordinates"], "latex": "d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}", "variables": [ { "symbol": "d", "meaning": "distance between the two points", "si_unit": null }, { "symbol": "x_1, y_1", "meaning": "coordinates of the first point", "si_unit": null }, { "symbol": "x_2, y_2", "meaning": "coordinates of the second point", "si_unit": null } ], "tags": ["geometry", "coordinates", "distance"], "description": "Gives the straight-line distance between two points in the Cartesian plane as an application of the Pythagorean theorem.", "source": "standard result; see e.g. Stewart, Redlin & Watson, Precalculus" }, { "name": "Slope of a line (two-point form)", "subject": "math", "topic": "coordinate geometry", "concepts": ["slope", "rate of change", "linear function"], "latex": "m = \\frac{y_2 - y_1}{x_2 - x_1}", "variables": [ { "symbol": "m", "meaning": "slope of the line through the two points", "si_unit": null }, { "symbol": "x_1, y_1", "meaning": "coordinates of the first point", "si_unit": null }, { "symbol": "x_2, y_2", "meaning": "coordinates of the second point, with x_2 not equal to x_1", "si_unit": null } ], "tags": ["algebra", "lines", "slope"], "description": "Computes the slope of a straight line as the ratio of the change in y to the change in x between two distinct points on it.", "source": "standard result; see e.g. Stewart, Redlin & Watson, Precalculus" }, { "name": "Binomial theorem", "subject": "math", "topic": "algebra and combinatorics", "concepts": ["binomial expansion", "binomial coefficient", "combinatorics"], "latex": "(x + y)^n = \\sum_{k=0}^{n} \\binom{n}{k} x^{\\,n-k} y^{\\,k}", "variables": [ { "symbol": "n", "meaning": "non-negative integer exponent", "si_unit": null }, { "symbol": "k", "meaning": "summation index running from 0 to n", "si_unit": null }, { "symbol": "\\binom{n}{k}", "meaning": "binomial coefficient n!/(k!(n-k)!)", "si_unit": null } ], "tags": ["algebra", "combinatorics", "expansion"], "description": "Expands a power of a binomial into a sum of terms whose coefficients are the binomial coefficients.", "source": "standard result; see e.g. Rosen, Discrete Mathematics and Its Applications" }, { "name": "Power rule for derivatives", "subject": "math", "topic": "differential calculus", "concepts": ["derivative", "power function", "differentiation rule"], "latex": "\\frac{d}{dx} x^{n} = n x^{\\,n-1}", "variables": [ { "symbol": "x", "meaning": "independent variable", "si_unit": null }, { "symbol": "n", "meaning": "real exponent of the power function", "si_unit": null } ], "tags": ["calculus", "derivative", "differentiation"], "description": "Differentiates a power function by multiplying by the exponent and lowering the exponent by one.", "source": "standard result; see e.g. Thomas' Calculus" }, { "name": "Fundamental theorem of calculus (evaluation part)", "subject": "math", "topic": "integral calculus", "concepts": ["definite integral", "antiderivative", "continuity"], "latex": "\\int_a^b f(x)\\,dx = F(b) - F(a), \\qquad F'(x) = f(x)", "variables": [ { "symbol": "f", "meaning": "continuous integrand on the interval [a, b]", "si_unit": null }, { "symbol": "F", "meaning": "any antiderivative of f on [a, b]", "si_unit": null }, { "symbol": "a, b", "meaning": "lower and upper limits of integration", "si_unit": null } ], "tags": ["calculus", "integral", "antiderivative"], "description": "Evaluates the definite integral of a continuous function as the difference of any antiderivative at the two limits of integration.", "source": "standard result; see e.g. Apostol, Calculus, Volume 1" }, { "name": "Sum of an arithmetic progression", "subject": "math", "topic": "sequences and series", "concepts": ["arithmetic progression", "common difference", "partial sum"], "latex": "S_n = \\frac{n}{2}\\bigl(2a + (n-1)d\\bigr)", "variables": [ { "symbol": "S_n", "meaning": "sum of the first n terms", "si_unit": null }, { "symbol": "n", "meaning": "number of terms summed", "si_unit": null }, { "symbol": "a", "meaning": "first term of the progression", "si_unit": null }, { "symbol": "d", "meaning": "common difference between consecutive terms", "si_unit": null } ], "tags": ["sequences", "series", "arithmetic progression"], "description": "Sums the first n terms of an arithmetic progression with first term a and common difference d.", "source": "standard result; see e.g. NCERT Mathematics Class XI, Sequences and Series" }, { "name": "Sum of a geometric progression", "subject": "math", "topic": "sequences and series", "concepts": ["geometric progression", "common ratio", "partial sum"], "latex": "S_n = a\\,\\frac{1 - r^{\\,n}}{1 - r}, \\qquad r \\neq 1", "variables": [ { "symbol": "S_n", "meaning": "sum of the first n terms", "si_unit": null }, { "symbol": "n", "meaning": "number of terms summed", "si_unit": null }, { "symbol": "a", "meaning": "first term of the progression", "si_unit": null }, { "symbol": "r", "meaning": "common ratio between consecutive terms", "si_unit": null } ], "tags": ["sequences", "series", "geometric progression"], "description": "Sums the first n terms of a geometric progression with first term a and common ratio r not equal to one.", "source": "standard result; see e.g. NCERT Mathematics Class XI, Sequences and Series" }, { "name": "Euler's identity", "subject": "math", "topic": "complex numbers", "concepts": ["complex exponential", "Euler's formula", "imaginary unit"], "latex": "e^{i\\pi} + 1 = 0", "variables": [ { "symbol": "e", "meaning": "base of the natural logarithm, approximately 2.71828", "si_unit": null }, { "symbol": "i", "meaning": "imaginary unit with i^2 = -1", "si_unit": null }, { "symbol": "\\pi", "meaning": "ratio of a circle's circumference to its diameter", "si_unit": null } ], "tags": ["complex numbers", "exponential", "identity"], "description": "The special case of Euler's formula at an angle of pi radians, linking the constants e, i, pi, 1 and 0 in one equation.", "source": "standard result; see e.g. Ahlfors, Complex Analysis" }, { "name": "Law of cosines", "subject": "math", "topic": "trigonometry", "concepts": ["triangle", "cosine", "generalised Pythagorean theorem"], "latex": "c^2 = a^2 + b^2 - 2ab\\cos C", "variables": [ { "symbol": "a, b", "meaning": "lengths of the two sides enclosing the angle C", "si_unit": null }, { "symbol": "c", "meaning": "length of the side opposite the angle C", "si_unit": null }, { "symbol": "C", "meaning": "interior angle between sides a and b", "si_unit": "rad" } ], "tags": ["trigonometry", "triangle", "geometry"], "description": "Relates the three sides of any triangle to the cosine of one of its angles, reducing to the Pythagorean theorem when that angle is a right angle.", "source": "standard result; see e.g. Stewart, Redlin & Watson, Precalculus" }, { "name": "Newton's second law of motion", "subject": "physics", "topic": "classical mechanics", "concepts": ["force", "mass", "acceleration", "dynamics"], "latex": "\\vec{F} = m\\vec{a}", "variables": [ { "symbol": "\\vec{F}", "meaning": "net force acting on the body", "si_unit": "N" }, { "symbol": "m", "meaning": "mass of the body", "si_unit": "kg" }, { "symbol": "\\vec{a}", "meaning": "acceleration of the body", "si_unit": "m/s^2" } ], "tags": ["mechanics", "force", "motion"], "description": "States that the acceleration of a body of constant mass is proportional to the net force on it and points in the same direction.", "source": "standard result; see e.g. Halliday, Resnick & Walker, Fundamentals of Physics" }, { "name": "Kinetic energy", "subject": "physics", "topic": "work and energy", "concepts": ["kinetic energy", "work-energy theorem", "motion"], "latex": "E_k = \\frac{1}{2} m v^2", "variables": [ { "symbol": "E_k", "meaning": "translational kinetic energy of the body", "si_unit": "J" }, { "symbol": "m", "meaning": "mass of the body", "si_unit": "kg" }, { "symbol": "v", "meaning": "speed of the body", "si_unit": "m/s" } ], "tags": ["mechanics", "energy", "motion"], "description": "Gives the energy a body of mass m carries because of its translational motion at speed v, valid for speeds well below that of light.", "source": "standard result; see e.g. Halliday, Resnick & Walker, Fundamentals of Physics" }, { "name": "Gravitational potential energy near the Earth's surface", "subject": "physics", "topic": "work and energy", "concepts": ["potential energy", "uniform gravitational field", "conservation of energy"], "latex": "U = mgh", "variables": [ { "symbol": "U", "meaning": "gravitational potential energy relative to the chosen reference height", "si_unit": "J" }, { "symbol": "m", "meaning": "mass of the body", "si_unit": "kg" }, { "symbol": "g", "meaning": "acceleration due to gravity, about 9.81 m/s^2 at the Earth's surface", "si_unit": "m/s^2" }, { "symbol": "h", "meaning": "height above the reference level", "si_unit": "m" } ], "tags": ["mechanics", "energy", "gravity"], "description": "Gives the stored energy of a mass raised a height h in a gravitational field that is uniform over that height.", "source": "standard result; see e.g. Halliday, Resnick & Walker, Fundamentals of Physics" }, { "name": "Ohm's law", "subject": "physics", "topic": "current electricity", "concepts": ["voltage", "current", "resistance", "ohmic conductor"], "latex": "V = IR", "variables": [ { "symbol": "V", "meaning": "potential difference across the conductor", "si_unit": "V" }, { "symbol": "I", "meaning": "current through the conductor", "si_unit": "A" }, { "symbol": "R", "meaning": "resistance of the conductor", "si_unit": "Ω" } ], "tags": ["electricity", "circuits", "resistance"], "description": "For an ohmic conductor at constant temperature, the current through it is proportional to the potential difference across it.", "source": "standard result; see e.g. Halliday, Resnick & Walker, Fundamentals of Physics" }, { "name": "Coulomb's law", "subject": "physics", "topic": "electrostatics", "concepts": ["electrostatic force", "point charge", "inverse-square law"], "latex": "F = \\frac{1}{4\\pi\\varepsilon_0}\\,\\frac{q_1 q_2}{r^2}", "variables": [ { "symbol": "F", "meaning": "magnitude of the electrostatic force between the charges", "si_unit": "N" }, { "symbol": "q_1, q_2", "meaning": "the two point charges", "si_unit": "C" }, { "symbol": "r", "meaning": "separation between the charges", "si_unit": "m" }, { "symbol": "\\varepsilon_0", "meaning": "electric constant (vacuum permittivity), about 8.854 × 10⁻¹² F/m", "si_unit": "F/m" } ], "tags": ["electrostatics", "force", "charge"], "description": "Gives the force between two point charges at rest in vacuum, acting along the line joining them and falling off as the inverse square of their separation.", "source": "standard result; see e.g. Griffiths, Introduction to Electrodynamics; constant from CODATA" }, { "name": "Period of a simple pendulum", "subject": "physics", "topic": "oscillations", "concepts": ["simple harmonic motion", "small-angle approximation", "period"], "latex": "T = 2\\pi\\sqrt{\\frac{L}{g}}", "variables": [ { "symbol": "T", "meaning": "period of one complete oscillation", "si_unit": "s" }, { "symbol": "L", "meaning": "length of the pendulum", "si_unit": "m" }, { "symbol": "g", "meaning": "acceleration due to gravity", "si_unit": "m/s^2" } ], "tags": ["oscillations", "pendulum", "harmonic motion"], "description": "Gives the period of an ideal simple pendulum in the small-angle approximation, where it is independent of the bob's mass and of the amplitude.", "source": "standard result; see e.g. Halliday, Resnick & Walker, Fundamentals of Physics" }, { "name": "de Broglie wavelength", "subject": "physics", "topic": "quantum physics", "concepts": ["wave-particle duality", "matter waves", "momentum"], "latex": "\\lambda = \\frac{h}{p}", "variables": [ { "symbol": "\\lambda", "meaning": "de Broglie wavelength of the particle", "si_unit": "m" }, { "symbol": "h", "meaning": "Planck constant, 6.626 × 10⁻³⁴ J·s", "si_unit": "J·s" }, { "symbol": "p", "meaning": "momentum of the particle", "si_unit": "kg·m/s" } ], "tags": ["quantum", "wave-particle duality", "momentum"], "description": "Assigns a wavelength to a moving particle that is inversely proportional to its momentum, the basis of wave-particle duality.", "source": "standard result; see e.g. Eisberg & Resnick, Quantum Physics; constant from CODATA" }, { "name": "Mass-energy equivalence", "subject": "physics", "topic": "special relativity", "concepts": ["rest energy", "mass", "speed of light"], "latex": "E = mc^2", "variables": [ { "symbol": "E", "meaning": "rest energy of the body", "si_unit": "J" }, { "symbol": "m", "meaning": "rest mass of the body", "si_unit": "kg" }, { "symbol": "c", "meaning": "speed of light in vacuum, exactly 299 792 458 m/s", "si_unit": "m/s" } ], "tags": ["relativity", "energy", "mass"], "description": "States that the rest energy of a body is its rest mass times the square of the speed of light in vacuum.", "source": "standard result; see e.g. Taylor & Wheeler, Spacetime Physics; constant from the SI definition" }, { "name": "Newton's law of universal gravitation", "subject": "physics", "topic": "gravitation", "concepts": ["gravitational force", "inverse-square law", "point mass"], "latex": "F = G\\,\\frac{m_1 m_2}{r^2}", "variables": [ { "symbol": "F", "meaning": "magnitude of the gravitational attraction between the masses", "si_unit": "N" }, { "symbol": "G", "meaning": "gravitational constant, about 6.674 × 10⁻¹¹ N·m²/kg²", "si_unit": "N·m^2/kg^2" }, { "symbol": "m_1, m_2", "meaning": "the two masses", "si_unit": "kg" }, { "symbol": "r", "meaning": "distance between the centres of the masses", "si_unit": "m" } ], "tags": ["gravitation", "force", "mechanics"], "description": "Every pair of point masses attracts along the line joining them with a force proportional to the product of the masses and inversely proportional to the square of their separation.", "source": "standard result; see e.g. Halliday, Resnick & Walker, Fundamentals of Physics; constant from CODATA" }, { "name": "Snell's law of refraction", "subject": "physics", "topic": "geometrical optics", "concepts": ["refraction", "refractive index", "angle of incidence"], "latex": "n_1 \\sin\\theta_1 = n_2 \\sin\\theta_2", "variables": [ { "symbol": "n_1, n_2", "meaning": "refractive indices of the first and second media", "si_unit": null }, { "symbol": "\\theta_1", "meaning": "angle of incidence, measured from the normal", "si_unit": "rad" }, { "symbol": "\\theta_2", "meaning": "angle of refraction, measured from the normal", "si_unit": "rad" } ], "tags": ["optics", "refraction", "light"], "description": "Relates the angles of incidence and refraction when light crosses the boundary between two media of different refractive index.", "source": "standard result; see e.g. Hecht, Optics" }, { "name": "Ideal gas law", "subject": "chemistry", "topic": "gas laws", "concepts": ["ideal gas", "state equation", "amount of substance"], "latex": "pV = nRT", "variables": [ { "symbol": "p", "meaning": "pressure", "si_unit": "Pa" }, { "symbol": "V", "meaning": "volume", "si_unit": "m^3" }, { "symbol": "n", "meaning": "amount of substance", "si_unit": "mol" }, { "symbol": "R", "meaning": "molar gas constant, 8.314 J·mol⁻¹·K⁻¹", "si_unit": "J/(mol·K)" }, { "symbol": "T", "meaning": "absolute temperature", "si_unit": "K" } ], "tags": ["gases", "thermodynamics"], "description": "Equation of state for an ideal gas relating pressure, volume, amount and temperature.", "source": "standard result; IUPAC Gold Book" }, { "name": "Molar concentration (molarity)", "subject": "chemistry", "topic": "solutions", "concepts": ["concentration", "amount of substance", "solution"], "latex": "c = \\frac{n}{V}", "variables": [ { "symbol": "c", "meaning": "amount concentration of the solute, commonly quoted in mol/L", "si_unit": "mol/m^3" }, { "symbol": "n", "meaning": "amount of substance of solute", "si_unit": "mol" }, { "symbol": "V", "meaning": "volume of the solution", "si_unit": "m^3" } ], "tags": ["solutions", "concentration", "stoichiometry"], "description": "Defines the amount concentration of a solute as the moles of solute divided by the total volume of the solution.", "source": "standard definition; IUPAC Gold Book, amount concentration" }, { "name": "pH definition", "subject": "chemistry", "topic": "acids and bases", "concepts": ["acidity", "hydrogen ion activity", "logarithmic scale"], "latex": "\\mathrm{pH} = -\\log_{10} a_{\\mathrm{H^{+}}}", "variables": [ { "symbol": "\\mathrm{pH}", "meaning": "acidity of the solution on a logarithmic scale", "si_unit": null }, { "symbol": "a_{\\mathrm{H^{+}}}", "meaning": "relative activity of the hydrogen ion, approximated by the numerical value of [H+] in mol/L for dilute solutions", "si_unit": null } ], "tags": ["acids", "bases", "equilibrium"], "description": "Defines pH as the negative base-ten logarithm of the hydrogen ion activity, so each unit of pH is a tenfold change in acidity.", "source": "standard definition; IUPAC Gold Book, pH" }, { "name": "Arrhenius equation", "subject": "chemistry", "topic": "chemical kinetics", "concepts": ["rate constant", "activation energy", "temperature dependence"], "latex": "k = A\\,e^{-E_a/(RT)}", "variables": [ { "symbol": "k", "meaning": "rate constant, whose units depend on the reaction order", "si_unit": null }, { "symbol": "A", "meaning": "pre-exponential factor, with the same units as k", "si_unit": null }, { "symbol": "E_a", "meaning": "activation energy of the reaction", "si_unit": "J/mol" }, { "symbol": "R", "meaning": "molar gas constant, 8.314 J·mol⁻¹·K⁻¹", "si_unit": "J/(mol·K)" }, { "symbol": "T", "meaning": "absolute temperature", "si_unit": "K" } ], "tags": ["kinetics", "activation energy", "temperature"], "description": "Describes how a reaction rate constant grows with temperature and falls with the activation energy of the reaction.", "source": "standard result; IUPAC Gold Book, Arrhenius equation; Atkins & de Paula, Physical Chemistry" }, { "name": "Gibbs free energy change", "subject": "chemistry", "topic": "chemical thermodynamics", "concepts": ["spontaneity", "enthalpy", "entropy", "free energy"], "latex": "\\Delta G = \\Delta H - T\\,\\Delta S", "variables": [ { "symbol": "\\Delta G", "meaning": "change in Gibbs free energy", "si_unit": "J/mol" }, { "symbol": "\\Delta H", "meaning": "change in enthalpy", "si_unit": "J/mol" }, { "symbol": "T", "meaning": "absolute temperature", "si_unit": "K" }, { "symbol": "\\Delta S", "meaning": "change in entropy", "si_unit": "J/(mol·K)" } ], "tags": ["thermodynamics", "energy", "spontaneity"], "description": "Combines enthalpy and entropy changes at a given temperature; a negative value means the process is spontaneous at constant temperature and pressure.", "source": "standard result; Atkins & de Paula, Physical Chemistry" }, { "name": "Nernst equation", "subject": "chemistry", "topic": "electrochemistry", "concepts": ["cell potential", "reaction quotient", "redox"], "latex": "E = E^{\\circ} - \\frac{RT}{zF}\\ln Q", "variables": [ { "symbol": "E", "meaning": "cell potential under the actual conditions", "si_unit": "V" }, { "symbol": "E^{\\circ}", "meaning": "standard cell potential", "si_unit": "V" }, { "symbol": "R", "meaning": "molar gas constant, 8.314 J·mol⁻¹·K⁻¹", "si_unit": "J/(mol·K)" }, { "symbol": "T", "meaning": "absolute temperature", "si_unit": "K" }, { "symbol": "z", "meaning": "number of electrons transferred in the cell reaction", "si_unit": null }, { "symbol": "F", "meaning": "Faraday constant, about 96485 C/mol", "si_unit": "C/mol" }, { "symbol": "Q", "meaning": "reaction quotient of the cell reaction", "si_unit": null } ], "tags": ["electrochemistry", "redox", "cell potential"], "description": "Corrects the standard electrode potential of a cell for non-standard concentrations and temperature through the reaction quotient.", "source": "standard result; IUPAC Gold Book, Nernst equation; Atkins & de Paula, Physical Chemistry" }, { "name": "First-order integrated rate law", "subject": "chemistry", "topic": "chemical kinetics", "concepts": ["reaction order", "rate constant", "exponential decay"], "latex": "\\ln[\\mathrm{A}]_t = \\ln[\\mathrm{A}]_0 - kt", "variables": [ { "symbol": "[\\mathrm{A}]_t", "meaning": "concentration of reactant A at time t", "si_unit": "mol/m^3" }, { "symbol": "[\\mathrm{A}]_0", "meaning": "initial concentration of reactant A", "si_unit": "mol/m^3" }, { "symbol": "k", "meaning": "first-order rate constant", "si_unit": "1/s" }, { "symbol": "t", "meaning": "elapsed reaction time", "si_unit": "s" } ], "tags": ["kinetics", "rate law", "first order"], "description": "Gives the concentration of a reactant that decays by a first-order reaction, so a plot of ln[A] against time is a straight line of slope -k.", "source": "standard result; Atkins & de Paula, Physical Chemistry" }, { "name": "Henderson-Hasselbalch equation", "subject": "chemistry", "topic": "acid-base equilibria", "concepts": ["buffer solution", "acid dissociation constant", "conjugate base"], "latex": "\\mathrm{pH} = \\mathrm{p}K_a + \\log_{10}\\frac{[\\mathrm{A^{-}}]}{[\\mathrm{HA}]}", "variables": [ { "symbol": "\\mathrm{pH}", "meaning": "pH of the buffer solution", "si_unit": null }, { "symbol": "\\mathrm{p}K_a", "meaning": "negative base-ten logarithm of the acid dissociation constant", "si_unit": null }, { "symbol": "[\\mathrm{A^{-}}]", "meaning": "concentration of the conjugate base", "si_unit": "mol/m^3" }, { "symbol": "[\\mathrm{HA}]", "meaning": "concentration of the undissociated weak acid", "si_unit": "mol/m^3" } ], "tags": ["acids", "buffers", "equilibrium"], "description": "Gives the pH of a buffer from the pKa of the weak acid and the ratio of conjugate base to acid, assuming both are present in appreciable amounts.", "source": "standard result; IUPAC Gold Book, Henderson-Hasselbalch equation" }, { "name": "Beer-Lambert law", "subject": "chemistry", "topic": "spectroscopy", "concepts": ["absorbance", "molar absorption coefficient", "path length"], "latex": "A = \\varepsilon\\,\\ell\\,c", "variables": [ { "symbol": "A", "meaning": "decadic absorbance of the sample", "si_unit": null }, { "symbol": "\\varepsilon", "meaning": "molar absorption coefficient at the measuring wavelength, commonly quoted in L·mol⁻¹·cm⁻¹", "si_unit": "m^2/mol" }, { "symbol": "\\ell", "meaning": "optical path length through the sample", "si_unit": "m" }, { "symbol": "c", "meaning": "amount concentration of the absorbing species", "si_unit": "mol/m^3" } ], "tags": ["spectroscopy", "absorbance", "analysis"], "description": "States that absorbance is proportional to the concentration of the absorbing species and to the path length through the sample, for dilute solutions and monochromatic light.", "source": "standard result; IUPAC Gold Book, Beer-Lambert law" }, { "name": "Equilibrium constant of a reaction", "subject": "chemistry", "topic": "chemical equilibrium", "concepts": ["equilibrium", "law of mass action", "reaction quotient"], "latex": "aA + bB \\rightleftharpoons cC + dD, \\qquad K_c = \\frac{[\\mathrm{C}]^{c}[\\mathrm{D}]^{d}}{[\\mathrm{A}]^{a}[\\mathrm{B}]^{b}}", "variables": [ { "symbol": "K_c", "meaning": "concentration equilibrium constant, dimensionless when concentrations are taken relative to the standard concentration", "si_unit": null }, { "symbol": "[\\mathrm{A}], [\\mathrm{B}]", "meaning": "equilibrium concentrations of the reactants", "si_unit": "mol/m^3" }, { "symbol": "[\\mathrm{C}], [\\mathrm{D}]", "meaning": "equilibrium concentrations of the products", "si_unit": "mol/m^3" }, { "symbol": "a, b, c, d", "meaning": "stoichiometric coefficients of the balanced equation", "si_unit": null } ], "tags": ["equilibrium", "mass action", "thermodynamics"], "description": "Expresses the law of mass action: at equilibrium the products of the species concentrations raised to their stoichiometric coefficients have a fixed ratio at a given temperature.", "source": "standard result; IUPAC Gold Book, equilibrium constant" }, { "name": "Bayes' theorem", "subject": "statistics", "topic": "conditional probability", "concepts": ["conditional probability", "prior", "posterior", "likelihood"], "latex": "P(A \\mid B) = \\frac{P(B \\mid A)\\,P(A)}{P(B)}", "variables": [ { "symbol": "P(A \\mid B)", "meaning": "posterior probability of A given B", "si_unit": null }, { "symbol": "P(B \\mid A)", "meaning": "likelihood of observing B when A holds", "si_unit": null }, { "symbol": "P(A)", "meaning": "prior probability of A", "si_unit": null }, { "symbol": "P(B)", "meaning": "marginal probability of the evidence B, which must be non-zero", "si_unit": null } ], "tags": ["probability", "inference", "bayes"], "description": "Relates the posterior probability of a hypothesis to its prior and the likelihood of the evidence.", "source": "standard result; see e.g. DeGroot & Schervish, Probability and Statistics" }, { "name": "Sample mean", "subject": "statistics", "topic": "descriptive statistics", "concepts": ["average", "central tendency", "estimator"], "latex": "\\bar{x} = \\frac{1}{n}\\sum_{i=1}^{n} x_i", "variables": [ { "symbol": "\\bar{x}", "meaning": "arithmetic mean of the sample", "si_unit": null }, { "symbol": "n", "meaning": "number of observations in the sample", "si_unit": null }, { "symbol": "x_i", "meaning": "the i-th observation", "si_unit": null } ], "tags": ["descriptive statistics", "average", "estimation"], "description": "The arithmetic average of a sample, an unbiased estimator of the population mean.", "source": "standard definition; see e.g. Casella & Berger, Statistical Inference" }, { "name": "Sample variance", "subject": "statistics", "topic": "descriptive statistics", "concepts": ["dispersion", "degrees of freedom", "unbiased estimator"], "latex": "s^2 = \\frac{1}{n-1}\\sum_{i=1}^{n}\\left(x_i - \\bar{x}\\right)^2", "variables": [ { "symbol": "s^2", "meaning": "sample variance", "si_unit": null }, { "symbol": "n", "meaning": "number of observations, with n greater than 1", "si_unit": null }, { "symbol": "x_i", "meaning": "the i-th observation", "si_unit": null }, { "symbol": "\\bar{x}", "meaning": "sample mean", "si_unit": null } ], "tags": ["descriptive statistics", "variance", "spread"], "description": "Measures the spread of a sample about its mean; dividing by n-1 rather than n makes it an unbiased estimator of the population variance.", "source": "standard definition; see e.g. Casella & Berger, Statistical Inference" }, { "name": "Standard error of the mean", "subject": "statistics", "topic": "sampling distributions", "concepts": ["sampling variability", "precision", "sample size"], "latex": "\\mathrm{SE}(\\bar{x}) = \\frac{s}{\\sqrt{n}}", "variables": [ { "symbol": "\\mathrm{SE}(\\bar{x})", "meaning": "estimated standard deviation of the sample mean", "si_unit": null }, { "symbol": "s", "meaning": "sample standard deviation", "si_unit": null }, { "symbol": "n", "meaning": "sample size", "si_unit": null } ], "tags": ["inference", "sampling", "standard error"], "description": "Estimates how much the sample mean would vary from sample to sample, shrinking with the square root of the sample size.", "source": "standard result; see e.g. DeGroot & Schervish, Probability and Statistics" }, { "name": "z-score (standard score)", "subject": "statistics", "topic": "standardisation", "concepts": ["standardisation", "normal distribution", "comparison of scales"], "latex": "z = \\frac{x - \\mu}{\\sigma}", "variables": [ { "symbol": "z", "meaning": "standard score of the observation", "si_unit": null }, { "symbol": "x", "meaning": "observed value", "si_unit": null }, { "symbol": "\\mu", "meaning": "population mean", "si_unit": null }, { "symbol": "\\sigma", "meaning": "population standard deviation", "si_unit": null } ], "tags": ["standardisation", "normal distribution", "inference"], "description": "Expresses an observation as the number of standard deviations it lies above or below the mean, putting different scales on common footing.", "source": "standard definition; see e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics" }, { "name": "Normal probability density function", "subject": "statistics", "topic": "continuous distributions", "concepts": ["normal distribution", "probability density", "bell curve"], "latex": "f(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}}\\exp\\!\\left(-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right)", "variables": [ { "symbol": "f(x)", "meaning": "probability density at the value x", "si_unit": null }, { "symbol": "\\mu", "meaning": "mean of the distribution", "si_unit": null }, { "symbol": "\\sigma", "meaning": "standard deviation of the distribution, which must be positive", "si_unit": null }, { "symbol": "x", "meaning": "value at which the density is evaluated", "si_unit": null } ], "tags": ["probability", "normal distribution", "density"], "description": "The density of the normal (Gaussian) distribution, the symmetric bell-shaped curve centred at the mean with spread set by the standard deviation.", "source": "standard result; see e.g. Casella & Berger, Statistical Inference" }, { "name": "Binomial probability mass function", "subject": "statistics", "topic": "discrete distributions", "concepts": ["Bernoulli trials", "binomial distribution", "counting successes"], "latex": "P(X = k) = \\binom{n}{k} p^{k}\\,(1-p)^{\\,n-k}", "variables": [ { "symbol": "P(X = k)", "meaning": "probability of exactly k successes", "si_unit": null }, { "symbol": "n", "meaning": "number of independent trials", "si_unit": null }, { "symbol": "k", "meaning": "number of successes, an integer from 0 to n", "si_unit": null }, { "symbol": "p", "meaning": "probability of success on a single trial", "si_unit": null } ], "tags": ["probability", "discrete distribution", "binomial"], "description": "Gives the probability of exactly k successes in n independent trials that each succeed with the same probability p.", "source": "standard result; see e.g. DeGroot & Schervish, Probability and Statistics" }, { "name": "Poisson probability mass function", "subject": "statistics", "topic": "discrete distributions", "concepts": ["Poisson distribution", "rare events", "rate parameter"], "latex": "P(X = k) = \\frac{\\lambda^{k} e^{-\\lambda}}{k!}", "variables": [ { "symbol": "P(X = k)", "meaning": "probability of exactly k events in the interval", "si_unit": null }, { "symbol": "\\lambda", "meaning": "expected number of events in the interval, which is also the variance", "si_unit": null }, { "symbol": "k", "meaning": "number of events, a non-negative integer", "si_unit": null } ], "tags": ["probability", "discrete distribution", "poisson"], "description": "Gives the probability of k events in a fixed interval when events occur independently at a constant average rate.", "source": "standard result; see e.g. DeGroot & Schervish, Probability and Statistics" }, { "name": "Pearson correlation coefficient", "subject": "statistics", "topic": "association between variables", "concepts": ["linear association", "covariance", "correlation"], "latex": "r = \\frac{\\sum_{i=1}^{n}(x_i - \\bar{x})(y_i - \\bar{y})}{\\sqrt{\\sum_{i=1}^{n}(x_i - \\bar{x})^2 \\sum_{i=1}^{n}(y_i - \\bar{y})^2}}", "variables": [ { "symbol": "r", "meaning": "sample correlation coefficient, always between -1 and 1", "si_unit": null }, { "symbol": "x_i, y_i", "meaning": "the i-th paired observations", "si_unit": null }, { "symbol": "\\bar{x}, \\bar{y}", "meaning": "sample means of x and y", "si_unit": null }, { "symbol": "n", "meaning": "number of paired observations", "si_unit": null } ], "tags": ["correlation", "association", "regression"], "description": "Measures the strength and direction of the linear association between two variables on a scale from -1 to 1; it does not imply causation.", "source": "standard definition; see e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics" }, { "name": "Simple linear regression slope (least squares)", "subject": "statistics", "topic": "regression", "concepts": ["least squares", "slope estimate", "linear model"], "latex": "\\hat{\\beta}_1 = \\frac{\\sum_{i=1}^{n}(x_i - \\bar{x})(y_i - \\bar{y})}{\\sum_{i=1}^{n}(x_i - \\bar{x})^2}", "variables": [ { "symbol": "\\hat{\\beta}_1", "meaning": "least-squares estimate of the slope", "si_unit": null }, { "symbol": "x_i, y_i", "meaning": "the i-th paired observations of predictor and response", "si_unit": null }, { "symbol": "\\bar{x}, \\bar{y}", "meaning": "sample means of the predictor and the response", "si_unit": null }, { "symbol": "n", "meaning": "number of paired observations", "si_unit": null } ], "tags": ["regression", "least squares", "linear model"], "description": "The ordinary least-squares estimate of the slope in a simple linear regression, equal to the sample covariance of x and y divided by the variance of x.", "source": "standard result; see e.g. Casella & Berger, Statistical Inference" } ]