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Bond Duration & DV01 formulas: Macaulay Duration, Modified Duration & rate shock
Calculate PV-weighted Macaulay Duration (years), interest-rate sensitivity Modified Duration, and DV01 dollar risk per 1 basis point (0.01%) yield shift.
Read the guide →Engineering & Fluid MechanicsBond’s Third Theory of Comminution Formula: Work Index (Wi) & Ball Mill Power
Calculate specific grinding energy W = 10 * Wi * (1/SQRT(P80) - 1/SQRT(F80)) in kWh/tonne and pinion shaft power (kW) from Fred C. Bond’s Work Index Wi and 80%-passing feed/product sizes in micrometers.
Read the guide →Engineering & ScienceBondi Spherical Accretion Radius (RB) & Black-Hole Gas Capture Formula
Calculate the gravitational capture Bondi radius RB = (2 * G * M) / c_s^2 in Astronomical Units (AU) and Bondi-Hoyle-Lyttleton radius R_BHL = (2 * G * M) / (c_s^2 + v_rel^2) for black holes and protostars.
Read the guide →Economics & Market MicrostructureBorda Count vs. Condorcet Winner Formula: Ranked Ballot Points & Pairwise Majority
Evaluate ranked-choice voter profiles by computing Jean-Charles de Borda’s positional score alongside Marquis de Condorcet’s pairwise head-to-head majority test—exposing when plurality, Borda, and Condorcet winners diverge!
Read the guide →Actuarial Science & Loss ReservingBornhuetter-Ferguson vs Chain-Ladder IBNR Reserve Formula
Compare Chain-Ladder IBNR = Paid * (CDF - 1) against Bornhuetter-Ferguson IBNR = EP * ELR * (1 - 1 / CDF) to stabilize immature accident years.
Read the guide →Semiconductor Foundry Economics, Defect Clustering & Die YieldBose-Einstein Multi-Mask Critical-Layer Wafer Yield Formula
Calculate Single-Mask Seeds Defect-Clustered Yield (%), n-Mask Bose-Einstein Compound Fab Yield (%), Integer Good Known Dies per Wafer, and Yield-Adjusted Die Cost Multiplier.
Read the guide →EngineeringBoussinesq Point-Load Vertical Soil Stress Increment & Influence Factor
Calculate foundation subsurface vertical stress increment Delta_sigma_z = (3 * Q / (2 * PI() * z^2)) * (1 / (1 + (r / z)^2))^2.5 and Boussinesq influence factor I_B at depth z and radial offset r.
Read the guide →Civil, Structural & Geotechnical EngineeringBouwer-Rice Unconfined Aquifer Slug-Test Formula: Hydraulic Conductivity K (m/day)
Determine in-situ unconfined or partially penetrating monitoring-well hydraulic conductivity K = (r_c^2 * LN(R_e / r_w)) / (2 * L_e * t) * LN(y_0 / y_t) using Herman Bouwer and R. C. Rice’s (1976) slug-recovery method.
Read the guide →Healthcare & Clinical CalculationsBova Score for Normotensive Acute Pulmonary Embolism (Stage I–III)
Stratify 30-day PE-related hemodynamic collapse risk across normotensive pulmonary embolism patients (SBP >= 90 mmHg) using Carlo Bova’s 0–7 score (Borderline SBP 90–100, Elevated Troponin, RV Dysfunction, Heart Rate >= 110).
Read the guide →Biostatistics & Clinical TrialsBowker Symmetry and Stuart-Maxwell Marginal Homogeneity on 3x3 Tables
Extend McNemar’s paired test to 3x3 ordinal rating tables using Bowker’s 3-df off-diagonal symmetry chi-square and Stuart-Maxwell’s 2-df marginal homogeneity quadratic form.
Read the guide →Charts & Data VisualizationBox-and-Whisker Plot Formula: Stacked Q1 Pedestal, Median Split & Whisker Lengths
Construct John Tukey’s 5-Number Summary Box-and-Whisker Plot from any Stacked Column chart using exact formulas for Hidden Q1 Pedestal, Lower Box (Median - Q1), Upper Box (Q3 - Median), and Lower/Upper Whisker Spans.
Read the guide →Healthcare & ClinicalBraden Scale for Predicting Pressure Ulcer / Injury Risk (6–23 Point Nursing Score)
Calculate the 6–23 point Braden Scale across Sensory Perception, Moisture, Activity, Mobility, Nutrition (1–4 each), and Friction & Shear (1–3) to trigger hospital pressure-injury prevention bundles.
Read the guide →Statistical & Clinical AnalysisBradley-Terry Paired-Comparison Formula: LLM Arena, Sports & Sensory Win Probability
Compute pairwise head-to-head win probabilities P(i beats j) = EXP(beta_i) / (EXP(beta_i) + EXP(beta_j)) and worth ratios EXP(beta_i - beta_j) using Ralph Bradley & Milton Terry’s 1952 model powering Chatbot Arena LLM leaderboards and NCAA/sports rankings.
Read the guide →Operations Research & Transportation NetworksBraess’s Paradox & Wardrop Equilibrium Latency Formula
Calculate pre-bridge vs post-bridge Wardrop Nash travel time on a 4-link diamond network and prove when adding a zero-latency shortcut increases EVERY commuter’s travel time!
Read the guide →StatisticsBray-Curtis Dissimilarity & Sorensen-Dice Ecological Similarity
Compare aligned nonnegative abundance vectors with Bray-Curtis dissimilarity and their binary presence/absence sets with Sorensen-Dice similarity, explicitly handling missing and empty samples.
Read the guide →Charts & Visual DataBreak Chart Axis Formula: Piecewise Outlier Compression & Visual Gap Scaling
Prevent a single 10x outlier bar from crushing normal data series to unreadable slivers by building a piecewise Broken Y-Axis helper series with explicit lower-band fidelity, break-gap buffer, and compressed upper-band slope.
Read the guide →Lookup & ReferenceBreak Tied Ranks & Display Sorted Leaderboards (Helper Key & SORTBY)
Build a deterministic leaderboard by revenue, margin and a unique stable ID. Use multi-key SORTBY or an integer rank helper without adding fragile decimal offsets to scores.
Read the guide →StatisticalBreak ties in RANK: assign unique 1..N ranks with RANK.EQ and COUNTIF
Add COUNTIF($B$2:B2, B2) - 1 to RANK.EQ so tied scores receive sequential ranks 1, 2, 3, 4 for clean Top-N lookups.
Read the guide →FinancialBreak-even units and contribution margin: solve for zero-loss and target profit volume
Calculate unit contribution margin, whole break-even units with ROUNDUP, and required volume for a target operating profit.
Read the guide →Mechanical & Structural EngineeringBredt-Batho Closed Thin-Walled Box Torsion and Shear Flow Formula
Calculate closed-tube shear flow q = T / (2*A_m), thinnest-wall peak shear stress, and Bredt’s torsion constant J_closed vs open-slit stiffness collapse.
Read the guide →Aerospace & Flight Mechanics EngineeringBreguet Aircraft Cruise Range and Parabolic Polar (L/D)_max Formula
Calculate maximum aerodynamic efficiency (L/D)_max = 0.5 * SQRT((PI() * AR * e) / C_D0) and jet Breguet cruise range R_km = (V_kmh / c_hr) * (L/D) * LN(W_init / W_final).
Read the guide →EngineeringBreguet Aircraft Range Equation & Lift-to-Drag (L/D) Glide Distance formulas
Explore an idealized constant-parameter jet cruise segment and a separate steady still-air glide calculation.
Read the guide →Psychometrics & Clinical EpidemiologyBrennan-Prediger Free-Marginal Kappa and PABAK Formula
Diagnose Feinstein and Cicchetti’s skewed-prevalence kappa paradox using Byrt’s Prevalence Index and Brennan-Prediger’s q-category free-marginal kappa_n.
Read the guide →Financial ModelingBrenner-Subrahmanyam ATM Implied Volatility & Straddle Breakeven Formula
Extract At-The-Money implied volatility directly without Goal Seek using Brenner and Subrahmanyam’s (1988) SQRT(2*PI()/T) * (C_ATM / S) approximation, plus straddle breakevens and daily 1-SD move.
Read the guide →Statistical & PsychometricsBreslow-Day / Woolf Homogeneity of Odds Ratios Test (Effect Modification Q)
Test whether stratified log-odds ratios share a common true effect BEFORE pooling Mantel-Haenszel or Woolf estimates using Q = Σ w_i * (y_i - y_bar)^2 against χ²(k - 1).
Read the guide →Brewing, Cider & Packaged Beverage Engineering, Del Vecchio Tunnel Pasteurization Units (PU)Brewery Tunnel Pasteurization Units (PU) Lethality Formula
Calculate Del Vecchio Instantaneous Pasteurization Rate 1.393^(T-60) (PU/min), Total Delivered PU, Minimum Hold Time for Target PU (min), and Spec Realization (%).
Read the guide →Fermentation Science, Brewing & Beverage QCBrewing Plato, Real Attenuation & Balling ABV Formula
Calculate Apparent Attenuation (P_o - P_a)/P_o, Carl Balling’s Real Extract RE = 0.1808*P_o + 0.8192*P_a, Real Degree of Fermentation (RDF), and Finished Beer Alcohol by Volume (% ABV).
Read the guide →StatisticalBrier Score & Binary Log-Loss (Cross-Entropy) probability calibration formulas
Evaluate probabilistic forecasts and ML classifiers using Brier Score AVERAGE((p - y)^2) and Binary Cross-Entropy Log-Loss -AVERAGE(y*LN(p) + (1-y)*LN(1-p)).
Read the guide →Healthcare & ClinicalBrighton Pediatric Early Warning Score (PEWS) Ward Escalation Formula
Calculate Alan Monaghan’s Brighton Pediatric Early Warning Score (0 to 13 points) summing Behavior (0–3), Cardiovascular (0–3), and Respiratory (0–3) domains plus +2 each for Q15-minute nebulizers and persistent post-operative vomiting.
Read the guide →Engineering & ScienceBrillouin Function B_J(x): Quantum Paramagnetic Magnetization Saturation
Calculate fractional quantum paramagnetic magnetization M / M_sat = B_J(x) for discrete angular momentum J across high-field/cryogenic regimes vs. linear Curie-law approximation.
Read the guide →Engineering & ScienceBrillouin Scattering Acoustic Phonon Frequency Shift (nu_B) in Optical Fiber
Calculate the Léon Brillouin inelastic acoustic-phonon frequency shift nu_B = ((2 * n * v_a) / lambda_0) * SIN(RADIANS(theta / 2)) in GHz for distributed fiber-optic BOTDR strain/temperature sensors and SBS power limits.
Read the guide →Operations, Logistics & EngineeringBrinch Hansen (1970) Shape & Depth Factors Formula: Rectangular Footing Bearing Capacity
Upgrade 2D strip-footing bearing capacity to real 3D rectangular and square pads (B <= L) embedded at depth D_f using Jørgen Brinch Hansen’s (1970) shape multipliers (s_q, s_gamma, s_c) and depth factor d_q.
Read the guide →Metallurgy, ASTM E10 Hardness Testing & Structural Steel InspectionBrinell Hardness (HBW) Spherical-Cap & Steel UTS Formula
Calculate ASTM E10 Tungsten-Carbide Brinell Hardness HBW, Naive Projected-Circle Error HB, Estimated Carbon/Alloy Steel Ultimate Tensile Strength UTS (MPa), and d/D Validity Ratio.
Read the guide →Quantitative FinanceBrinson-Fachler Portfolio Attribution: Allocation, Selection & Interaction
Decompose active portfolio excess return into exact Brinson-Fachler Asset Allocation, Security Selection, and Interaction effects across sectors using SUMPRODUCT.
Read the guide →Venture Capital & Startup Cap TablesBroad-Based Weighted Average vs Full Ratchet Formula
Calculate Down-Round Preferred Conversion Price under Broad-Based Weighted Average CP_new = CP_0 * (A + B) / (A + C), Narrow-Based Weighted Average, and Full Ratchet anti-dilution.
Read the guide →Engineering & ScienceBroad-Crested Weir Discharge & Critical Crest Depth (y_c = 2/3 H_0) Formula
Calculate dam spillway and levee overflow discharge Q = C_d * (2/3)^1.5 * SQRT(g) * b * H_0^1.5 and critical crest flow depth y_c = (2/3) * H_0.
Read the guide →Marketing Mix Modeling & EconometricsBroadbent Adstock Decay & Hill Saturation Formula
Model marketing carryover A_t = X_t + lambda * A_{t-1}, adstock half-life -LN(2) / LN(lambda), steady-state stock X / (1 - lambda), and Hill S-curve diminishing returns.
Read the guide →Engineering & ScienceBrokaw Gas Mixture Viscosity Formula: Polar & Nonpolar Binary Interaction
Calculate low-pressure polar gas mixture viscosity μ_m = y_1*μ_1/(y_1 + y_2*φ_12) + y_2*μ_2/(y_2 + y_1*φ_21) using Brokaw dipole-corrected interaction factors.
Read the guide →Engineering & Fluid MechanicsBromley Film Boiling Correlation Formula: Horizontal Tube Vapor-Blanket Heat Transfer
Calculate stable film-boiling convective heat transfer above the Leidenfrost point on horizontal tubes using LeRoy A. Bromley’s 1950 Nusselt-analogue equation h_co = 0.62 * [k_v³ * rho_v * (rho_l - rho_v) * g * (h_fg + 0.4 * c_pv * deltaT_e) / (mu_v * D * deltaT_e)]^0.25.
Read the guide →Civil, Structural & Geotechnical EngineeringBroms Lateral Pile Capacity Formula: Short Rigid Free-Head Piles in Sand (Cohesionless)
Calculate ultimate lateral soil resistance H_ult (kN) and maximum pile bending moment M_max (kN·m) for short rigid free-head piles in cohesionless sand using Bengt B. Broms’s (1964, ASCE) 3 * K_p passive wedge solution.
Read the guide →Engineering & ScienceBrooks-Herring Screened Ionized Impurity Scattering Factor Formula
Compute the Debye-screened Brooks-Herring parameter b = 1.2925E14 * ε_r * (m*/m_0) * T^2 / n and Coulomb scattering function f_BH(b) = LN(1 + b) - b / (1 + b) in doped semiconductors.
Read the guide →Engineering & ScienceBrunauer-Emmett-Teller (BET) Multilayer Adsorption & Specific Surface Area
Calculate multilayer nitrogen physisorption uptake v = (v_m * c * x) / ((1 - x) * (1 + (c - 1) * x)) (cm^3 STP/g) across relative pressure x = P / P_0 and specific BET catalyst surface area S_BET (m^2/g).
Read the guide →Nonparametric Statistics & Clinical TrialsBrunner-Munzel Heteroscedastic Rank Test and Stochastic Superiority
Solve the nonparametric Behrens-Fisher unequal-variance problem using Brunner and Munzel’s (2000) placement-variance rank statistic W_BM and Satterthwaite degrees of freedom.
Read the guide →Charts & Data VisualizationBubble Chart Size Formula: Area-Proportional Radius SQRT(Z / Z_max) vs Diameter Distortion
Encode a third quantitative variable Z accurately in Bubble Charts by scaling circle radius as R_i = R_max * SQRT(Z_i / MAX(Z)) so visual disk area PI * R_i^2 stays strictly proportional to magnitude Z_i.
Read the guide →Public Economics & Club GoodsBuchanan Club Goods Optimal Membership Size Formula
Calculate James Buchanan’s Optimal Club Membership N* = SQRT(F / c_cong), Optimal Cost-Sharing Dues F / N*, Marginal Congestion Externality, and Per-Member Net Club Surplus.
Read the guide →Vertical Bulk Material Handling, Grain Legs, Cement Bucket Elevators & CEMA SizingBucket Elevator Critical RPM, Pole Distance & Capacity Formula
Calculate Centrifugal Pole Distance hp = g / omega² (m), Critical Zero-Throw Pulley Speed Ncrit (RPM), Linear Belt/Chain Speed (m/s), and Mass Elevating Capacity Qm (t/h).
Read the guide →Actuarial Science & Credibility PricingBühlmann-Straub Credibility Factor Z and Pure Premium Formula
Blend policyholder loss experience X_bar with manual class rate mu using Bühlmann-Straub credibility weight Z = (m * a) / (m * a + v) without division-by-zero traps.
Read the guide →Statistical & Conditional CountingBuild a Dynamic Two-Way Count Matrix (Mixed-Reference COUNTIFS Cross-Tab)
Cross-tabulate ticket or order counts by Priority (rows) and Channel (columns) using mixed-reference COUNTIFS that updates instantly without manual PivotTable refreshes.
Read the guide →Financial ModelingBuild a Formula-Driven KPI Scorecard Panel (Polarity-Aware Attainment, Weights & RAG Status)
Combine higher-is-better growth metrics and lower-is-better payback/defect metrics into one normalized executive KPI scorecard with weighted attainment and Red/Amber/Green status.
Read the guide →Logical & ConditionalBuild an "IF NOT ERROR" Rule (Execute Custom Logic Only When Lookup Succeeds)
Invert Excel’s built-in IFERROR—which only replaces errors while passing raw successes straight through—by combining IF(ISNA(XMATCH(...)), Fallback, Custom_Success_Result) or IF(NOT(ISERROR(expr)), Success, Error).
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