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Bachelier Normal ATM Option Price & Straddle Breakeven Formula
Price at-the-money options on interest rates, spreads, and commodities that can trade near or below zero using Bachelier’s (1900) arithmetic normal formula C_ATM = DF * sigma_N * SQRT(T / (2 * PI())).
Read the guide →Civil, Structural & Geotechnical EngineeringBagnold Stream Power Formula: Cross-Sectional Omega (W/m) & Unit Stream Power omega (W/m^2)
Quantify river geomorphic energy dissipation and channel incision/avulsion potential using Ralph A. Bagnold’s (1966, USGS PP 422-I) Cross-Sectional Stream Power Omega = rho * g * Q * S (W/m) and Specific Unit Stream Power omega = Omega / W (W/m^2).
Read the guide →Commercial Baking & Cereal Chemistry, Baker’s Percentages, Hydration & Strike Water DDTBaker’s Percentage Dough Scaling & Strike Water Temp Formula
Calculate Total Baker’s Formula Percentage (>100%), Scaled Total Flour Mass (kg), Hydration Water Mass (kg), and 3-Factor Strike Water Temperature for Desired Dough Temp (°C).
Read the guide →International Macroeconomics & DevelopmentBalassa-Samuelson Real Exchange Rate Appreciation Formula
Quantify trend real exchange rate appreciation (1 - gamma_T) * ((g_AT - g_AN) - (g_AT* - g_AN*)) and inflation differentials in catching-up economies.
Read the guide →FinancialBalloon loan formula: monthly payment (PMT with FV) & 5/25 balloon payoff balance
Model commercial balloon mortgages both ways: solve monthly PMT for a fixed maturity residual OR compute the 5-year balloon payoff on a 25-year amortization schedule.
Read the guide →Psychometrics & Clinical EpidemiologyBangdiwala’s B Observer Agreement Area Index Formula
Quantify k x k observer agreement geometrically in Excel and Arte Sheets via Bangdiwala’s B = SUM(n_ii^2) / SUM(R_i * C_i) and linear partial-credit weighted B_w.
Read the guide →Commercial Banking & Financial InstitutionsBank NIM, Efficiency Ratio & Basel III LCR Formula
Calculate bank Net Interest Margin NIM = (Interest_Inc - Interest_Exp) / Avg_Earning_Assets, Overhead Efficiency Ratio, and Basel III 30-day Liquidity Coverage Ratio LCR.
Read the guide →FinancialBank statement reconciliation formula: Adjusted Bank vs Adjusted Book balance
Reconcile month-end bank statements against general-ledger cash by adjusting for deposits in transit, outstanding checks, fees, interest, and NSF items.
Read the guide →Economics & Market MicrostructureBanzhaf Voting Power Index Formula: Weighted Quota, Critical Swings & Power vs. Seat Share
Expose why nominal shareholdings mislead in weighted board voting [Quota q; w_1, w_2, w_3] by counting critical swing coalitions and computing the Normalized Banzhaf Power Index.
Read the guide →Charts & Visual DataBar of Pie & Pie of Pie Chart Formula: Tail-Slice Threshold, "Other" Wedge & Sub-Bar Share
Build a clean Bar-of-Pie or Pie-of-Pie breakdown table by aggregating micro-slices below a percentage threshold into a single "Other" main-pie wedge and computing both within-"Other" and grand-total shares for the secondary bar stack.
Read the guide →Fixed Income & Portfolio AnalyticsBarbell vs Bullet Bond Convexity Advantage Formula
Match modified duration via w_short = (D_long - D_bullet)/(D_long - D_short) and calculate the Barbell Convexity Advantage and parallel-shift outperformance 0.5 * Delta_C * (Delta_y)^2.
Read the guide →Healthcare Operations & Clinical AnalyticsBarber-Johnson Hospital Bed Occupancy, TOI & ALOS Formula
Calculate all four Barber and Johnson (1973) inpatient hospital utilization metrics — Percentage Bed Occupancy, Average Length of Stay (ALOS), Turnover Interval (TOI), and Bed Throughput.
Read the guide →Healthcare & Clinical CalculationsBARD Score (BMI, AST/ALT Ratio, Diabetes) for MASLD Liver Fibrosis
Rule out advanced F3–F4 hepatic bridging fibrosis in NAFLD/MASLD without platelet counts using Stephen Harrison’s 0–4 BARD Score (BMI >= 28 [+1], AST/ALT >= 0.8 [+2], Type 2 Diabetes [+1]; 0–1 has 96% NPV).
Read the guide →EngineeringBarometric Altitude Pressure & ISA Tropospheric Lapse Rate formulas
Compute International Standard Atmosphere (ISA) air pressure P(h) = 101.325 * (1 - 2.25577E-5 * h)^5.25588 in kPa and standard temperature from 0 to 11,000 meters.
Read the guide →Quantitative Finance & DerivativesBarone-Adesi and Whaley Quadratic American Option Early-Exercise Premium
Calculate the Barone-Adesi-Whaley (1987) quadratic exponent q2, smooth-pasting coefficient A2, and early-exercise premium for American options with carry b < r.
Read the guide →Food, Pharma Blister & Flexible Packaging Engineering, ASTM F1249 WVTR Shelf-Life Mass BalanceBarrier Film WVTR Moisture Permeation & Shelf-Life Formula
Calculate Daily Water-Vapor Permeation Rate (g/day), Allowable Dry-Basis Moisture Budget (g), Predicted Crispness/Potency Shelf Life (days), and Shelf Life in Months.
Read the guide →Monetary Economics & Central BankingBarro-Gordon Inflation Bias & Rogoff Central Banker Formula
Calculate discretionary time-inconsistency inflation bias lambda * alpha * Delta_y*, excess social loss, and Rogoff conservative central-banker delegation.
Read the guide →Ground Improvement, Soft Marine Clay & Prefabricated Wick DrainsBarron Radial Consolidation Vertical Wick Drain Time Formula
Calculate Drain Spacing Ratio n = d_e / d_w, Exact Barron Geometric Factor F(n), Radial Consolidation Time Factor T_r, and Surcharge Waiting Time to Target Consolidation (Months).
Read the guide →Healthcare & ClinicalBarthel Index of Activities of Daily Living (0–100 ADL Stroke & Rehab Score)
Sum the 10-item Mahoney & Barthel Index (0–100 in 5-point steps across Feeding, Bathing, Grooming, Dressing, Bowels, Bladder, Toilet Use, Chair/Bed Transfers, Mobility, and Stairs) and classify functional dependency.
Read the guide →Healthcare & ClinicalBASDAI (Bath Ankylosing Spondylitis Disease Activity Index) Formula
Calculate the 0–10 Bath Ankylosing Spondylitis Disease Activity Index BASDAI = (Q1 + Q2 + Q3 + Q4 + (Q5 + Q6)/2) / 5 averaging morning stiffness severity and duration into a single domain.
Read the guide →StatisticalBaseball OPS (On-Base Plus Slugging), OBP, SLG & Isolated Power (ISO)
Calculate Major League Baseball On-Base Percentage (H + BB + HBP) / (AB + BB + HBP + SF), Total Bases shortcut H + 2B + 2*3B + 3*HR, Slugging, OPS, and Isolated Power.
Read the guide →Commercial Banking & Prudential RiskBasel CET1 Ratio, RWA Density & Texas Ratio Formula
Audit bank solvency from three angles: Basel III Common Equity Tier 1 Ratio CET1 / RWA, Risk-Weight Density RWA / Total_Assets, and Gerard Cassidy’s Non-Performing Asset Texas Ratio.
Read the guide →Banking Regulation & Asset-Liability ManagementBasel III Net Stable Funding Ratio (NSFR) Formula
Calculate Basel III Available Stable Funding (ASF), Required Stable Funding (RSF), Structural 1-Year Net Stable Funding Ratio NSFR = ASF / RSF (>= 100%), and Stable Funding Surplus/Deficit.
Read the guide →Marketing Analytics & ForecastingBass Diffusion Model Adoption Curve and Peak Time Formula
Forecast cumulative adopters N(t), peak sales timing t* = MAX(0, LN(q / p) / (p + q)), and peak adoption rate s(t*) = m * (p + q)^2 / (4 * q) using Frank Bass’s (1969) innovation-imitation model.
Read the guide →Grid Energy Storage (BESS) & ElectrochemistryBattery Coulombic, Voltaic & Round-Trip RTE Formula
Decompose Grid Battery Storage Round-Trip Energy Efficiency RTE = eta_Coulombic * eta_Voltaic from Charge/Discharge Amp-Hours and Mean Voltages, and quantify Dissipated Cycle Energy Loss.
Read the guide →Energy Finance & Grid Storage AnalyticsBattery Levelized Cost of Storage (LCOS) Formula
Calculate utility-scale battery LCOS ($/MWh discharged) using Capital Recovery Factor r/(1 - (1 + r)^(-N)), usable DoD cycles, and round-trip charging cost P_charge / eta_rt.
Read the guide →Electrochemical Energy Storage & EV PowertrainsBattery Ragone Specific Energy & Peak Power Formula
Calculate Matched-Load Peak Specific Power P_max = V_oc^2 / (4 * R_int * Mass), Ragone Voltage Efficiency (1 + SQRT(1 - P_load / P_max)) / 2, Delivered Specific Energy, and High-Power Runtime.
Read the guide →Treasury & Corporate FinanceBaumol-Tobin Optimal Cash Replenishment Size and Opportunity Cost Audit
Balance fixed securities-liquidation fees against foregone short-term yield using the Baumol-Tobin square-root cash model C* = SQRT(2 * b * T / i) and verify cost symmetry.
Read the guide →Economics & Market MicrostructureBaumol’s Cost Disease Formula: Relative Unit Labor Cost & Stagnant Sector GDP Share
Model William Baumol’s (1967) Unbalanced Growth Law: explain why healthcare, education, and live-service unit costs compound at ((1 + g_p) / (1 + g_s))^t when economy-wide wages track progressive-sector productivity.
Read the guide →Clinical Medicine & PhysiologyBaveno VII Portal Hypertension (CSPH) Rule-Of-5 Criteria & ANTICIPATE Probability Formula
Stratify compensated advanced chronic liver disease (cACLD) patients for Clinically Significant Portal Hypertension (CSPH, HVPG >= 10 mmHg) and endoscopic screening avoidance using Baveno VII (2022) Transient Elastography LSM + Platelet criteria and the ANTICIPATE logistic score.
Read the guide →StatisticalBayes' Theorem formula: Positive Predictive Value (PPV) & false-positive paradox
Use a hypothetical fraud detector to calculate predictive values from prevalence, sensitivity and specificity.
Read the guide →Law & Economics & Regulatory ComplianceBecker Optimal Enforcement Audit & Sanction Formula
Compute Becker / Polinsky-Shavell maximal-fine audit probability p* = H/W_max - (c_p*G_max)/W_max^2 and explain optimal under-deterrence when auditing is costly.
Read the guide →Healthcare & Clinical CalculationsBedside Schwartz Pediatric eGFR Formula (2009 IDMS Creatinine)
Calculate pediatric estimated glomerular filtration rate eGFR = (0.413 * Height_cm) / Serum_Cr_mgdL (mL/min/1.73 m^2) for children aged 1–17 years using IDMS-standardized enzymatic serum creatinine.
Read the guide →Brewing Packaging, Bright-Tank Cellaring & Draught Dispense, ASBC Zahm-Nagel CarbonationBeer Carbonation Volumes CO2 (Zahm-Nagel) & Line Formula
Calculate ASBC Equilibrium Dissolved Carbonation (Volumes CO2 and g/L), Draught Choker Restriction Line Length (ft), and Beer Style Carbonation Band from Gauge Pressure & Temp.
Read the guide →EngineeringBeer-Lambert Law formula: Absorbance (2 - LOG10(%T)) & Molar Concentration
Convert UV-Vis % Transmittance into Optical Absorbance A = 2 - LOG10(%T) and calculate unknown analyte concentration c = A / (ε · l).
Read the guide →Statistical & Clinical AnalysisBegg & Mazumdar Rank Correlation Test: Kendall’s τ Funnel Asymmetry Z-Score
Test meta-analytic publication bias nonparametrically using Colin Begg and Madhuchhanda Mazumdar’s (1994) Kendall rank correlation τ and exact variance σ_S² = k*(k - 1)*(2*k + 5) / 18.
Read the guide →Chemical, Process & Catalysis EngineeringBeggs-Brill Two-Phase Flow Formula: Mixture Froude Number & Horizontal Liquid Holdup H_L(0)
Determine horizontal gas-liquid pipeline Mixture Froude Number N_Fr = v_m^2 / (g * D), Input No-Slip Liquid Fraction lambda_L, and Segregated/Intermittent/Distributed Liquid Holdup H_L(0) via H. Dale Beggs and James P. Brill (1973).
Read the guide →Statistical & Clinical AnalysisBeighton Hypermobility Score Formula: 2017 EDS Age-Stratified Cutoffs (0 to 9)
Calculate the 9-point Beighton Joint Hypermobility Score and apply the 2017 International Ehlers-Danlos Syndrome (hEDS) age-adjusted generalized hypermobility thresholds (>= 6 prepubertal, >= 5 ages 13–50, >= 4 over 50).
Read the guide →Engineering & ScienceBélanger Equation: Hydraulic Jump Sequent Depth & Specific Energy Head Loss
Calculate rectangular stilling-basin sequent depth y_2 = 0.5*y_1*(SQRT(1 + 8*Fr_1^2) - 1) and turbulent energy dissipation ΔE = (y_2 - y_1)^3 / (4*y_1*y_2).
Read the guide →Engineering & Fluid MechanicsBell-Delaware Method Formula: Shell-Side Heat Transfer Coefficient (Jc, Jl, Jb, Js, Jr)
Correct ideal crossflow tube-bank heat transfer h_id for real TEMA shell-and-tube baffle geometry, leakage, bundle bypass, unequal spacing, and laminar gradients via h_s = h_id * Jc * Jl * Jb * Js * Jr.
Read the guide →Mechanical Machine Design, DIN 2092 Belleville Springs & Live LoadingBelleville Disc-Spring Parallel & Series Stack Formula
Calculate DIN 2092 Belleville Coned-Disc Spring Stack Load Capacity (N), Total Axial Deflection (mm), Effective Stack Stiffness Rate (N/mm), and Elastic Energy Stored (J).
Read the guide →Aviation Yield Management & Dynamic PricingBelobaba EMSR-a Nested Seat Protection Formula
Calculate Pairwise Littlewood Seat Protections y_13 and y_23 under Exponential High-Fare Demand, Belobaba’s Additive EMSR-a Joint Protection Y_12, and Nested Discount Class Booking Limit.
Read the guide →Bulk Material Handling, CEMA Belt Conveyors, Mining & Aggregate Plant EngineeringBelt Conveyor Bulk Tonnage Capacity & Drive Power Formula
Calculate CEMA Mass Conveying Capacity Q_m (t/h), Potential Gravity Lift Power P_lift (kW), Horizontal Idler Rolling Friction Power P_fric (kW), and Required Drive Motor Shaft Power (kW).
Read the guide →Open-Pit Mining Engineering, Quarry Drill-and-Blast Design & Fragmentation EconomicsBench Blasting Burden, Stemming & Powder Factor Formula
Calculate In-Situ Bank Rock Volume Per Hole (m³), Bulk Explosive Charge Mass Per Blasthole (kg), Volumetric Powder Factor (kg/m³), and Mass Powder Factor (kg/metric ton).
Read the guide →FinancialBeneish M-Score formula: 8-variable forensic earnings manipulation screen
Calculate an eight-input Beneish screening score from prepared accounting indices and compare its unrounded value with an explicit review threshold.
Read the guide →StatisticalBenford first-digit probabilities and significant-digit extraction
Extract significant digits from numeric amounts and compare a sample count with a theoretical Benford probability; deviations alone do not establish fraud.
Read the guide →Biostatistics & Multiple TestingBenjamini-Hochberg FDR vs Holm-Bonferroni Threshold Formula
Compare Benjamini-Hochberg step-up False Discovery Rate critical values (i/m)*q against Holm-Bonferroni step-down Family-Wise Error Rate cutoffs alpha/(m - i + 1) across sorted p-values.
Read the guide →Healthcare & Clinical FormulasBerg Balance Scale (BBS): 14-Item Functional Score & <45 Fall Cutoff
Sum all 14 ordinal (0–4) Berg Balance Scale tasks into total BBS (0–56), evaluate the < 45 multiple-fall threshold, and assign mobility assistance tiers (0–20 Wheelchair, 21–40 Assisted Ambulation, 41–56 Independent).
Read the guide →Healthcare & Clinical FormulasBerlin ARDS Severity (PaO2/FiO2) & Ventilator Oxygenation Index (OI)
Classify Acute Respiratory Distress Syndrome into Mild (200 < P/F <= 300), Moderate (100 < P/F <= 200), or Severe (P/F <= 100 on PEEP >= 5 cmH2O) per the 2012 Berlin Definition alongside Mean-Airway-Pressure Oxygenation Index OI = (100 * FiO2 * MAP) / PaO2.
Read the guide →EngineeringBernoulli's Equation & Venturi Flowmeter differential-pressure formula
Calculate volumetric flow rate from Venturi throat pressure drop via Q = Cd * A2 * SQRT(2 * dP / (rho * (1 - (A2/A1)^2))) and throat velocity V2 = Q / A2.
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