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Sharp and Fuzzy Regression Discontinuity (RDD) Local Jump Formula
Estimate local linear Regression Discontinuity cutoff intercepts INTERCEPT(Y, X - c) from both sides of threshold c and compute the Fuzzy RDD jump ratio tau_FRD.
Read the guide →FinancialSharpe Ratio formula: periodic and annualized risk-adjusted return
Measure excess return per unit of volatility with (AVERAGE(Returns) - RiskFree) / STDEV.S(Returns) and scale periodic Sharpe by SQRT(PeriodsPerYear).
Read the guide →FinancialSharpe ratio standard error, illustrative track-record threshold and serial-correlation scaling
Calculate an IID-normal large-sample Sharpe standard error and compare annual scaling under a stated AR(1) correlation assumption.
Read the guide →Sheet-Metal CAD/CAM Unfolding, Press-Brake Flat Patterns & Precision FabricationSheet-Metal Bend Allowance, K-Factor & Bend Deduction Formula
Calculate Neutral-Axis Arc Bend Allowance BA (mm), Outside Setback OSSB (mm), Bend Deduction BD = 2*OSSB - BA (mm), and Laser-Cut Flat Blank Length (mm).
Read the guide →Sheet-Metal Stamping & Press-Brake Forming, Gardiner Elastic-Plastic Springback LawSheet-Metal Bending Elastic Springback & Overbend Formula
Calculate Gardiner Springback Factor Ks, Unloaded Inside Bend Radius Rf (mm), Required Press-Brake Overbend Angle (deg), and Elastic Springback Angle Recovery (deg).
Read the guide →Progressive Stamping Dies, Sheet-Metal Blanking/Piercing & Turret PunchingSheet-Metal Blanking Shear Force & Die Clearance Formula
Calculate Single-Side Punch-to-Die Radial Clearance c (mm), Flat-Punch Peak Shear Force F_shear (kN), Stripper-Plate Unloading Force F_strip (kN), and Total Stamping Press Force (kN).
Read the guide →Engineering & ScienceSherwood Number (Sh), Schmidt Number (Sc) & Frössling Sphere Mass Transfer
Calculate the dimensionless Schmidt number Sc = nu / D_AB, Frössling-Ranz-Marshall sphere Sherwood number Sh = 2 + 0.6 * (Re^0.5) * (Sc^(1/3)), and convective mass-transfer coefficient k_c = (Sh * D_AB) / d_p.
Read the guide →ScienceShields Parameter for Incipient River Sediment Motion & Bed Shear
Calculate river bed shear stress tau_0 = rho * g * R_h * S, dimensionless Shields Stress tau_* = (rho * R_h * S) / ((rho_s - rho) * d_50), and gravel bedload mobilization status.
Read the guide →Marketing Science & Subscription SaaS AnalyticsShifted-Beta-Geometric (sBG) Cohort Retention Formula
Project contractual subscription survival S(t) = S(t - 1) * (beta + t - 1) / (alpha + beta + t - 1) under unobserved churn heterogeneity and explain rising cohort retention rates.
Read the guide →Quantitative Forecasting & Prediction MarketsShin Method vs Multiplicative Vig-Removal Formula
Strip bookmaker overround S = 1/O_1 + 1/O_2 - 1 via Hyun Song Shin’s (1993) insider-trading equilibrium to correct Favorite-Longshot Bias in implied probabilities.
Read the guide →Naval Architecture, Hydrostatic Stability & Maritime SafetyShip Metacentric Height (GM), Transverse BM & Righting Arm GZ
Calculate Vessel Transverse Metacentric Radius BM = I_T / Disp_Vol, Free-Surface-Corrected Fluid Metacentric Height GM_eff = KB + BM - KG - FSC, Small-Angle Righting Lever GZ = GM_eff*SIN(theta), and IMO Intact Stability Compliance.
Read the guide →Merchant Marine Cargo Operations, Plimsoll Load Lines & HydrostaticsShip Tons Per Centimeter (TPC) & Freshwater Allowance Formula
Calculate Saltwater Tons Per Centimeter Immersion TPC (t/cm), Plimsoll Freshwater Allowance FWA (mm), Brackish Dock Water Allowance DWA (mm), and Extra Permissible Cargo Mass (t).
Read the guide →StatisticalShock Index (HR / SBP), Modified Shock Index (HR / MAP) & Age Shock Index
Calculate heart-rate/blood-pressure ratios and an age-weighted index from consistently measured adult vital signs, without automatic diagnostic classification.
Read the guide →Semiconductor Physics, Photovoltaics & Tandem Solar Cell EngineeringShockley-Queisser Bandgap Thermalization & Voc Loss Formula
Calculate Solar Cell Sub-Bandgap Transmission vs Hot-Carrier Phonon Thermalization Loss (eV), Detailed-Balance Open-Circuit Voltage Voc (V), Extracted Electrical Work Per Photon (eV), and Monochromatic Photon Conversion Efficiency.
Read the guide →Aerospace SAE J2277 / AMS 2430 Shot Peening, Compressive Residual Stress & FatigueShot Peening Surface Coverage & Pass Count Formula
Calculate Multi-Pass Peened Dimple Coverage Cn = 1 - (1 - C1)^n (%), Residual Unpeened Parent Surface (%), Exact Continuous Passes for 98% Full Coverage, and Required Integer Nozzle Passes.
Read the guide →Engineering & ScienceShubnikov-de Haas (SdH) Onsager Frequency to 2DEG Carrier Density Formula
Extract two-dimensional electron gas (2DEG) sheet carrier density n_2D = 0.2418 * g_s * g_v * B_F (× 10^11 cm^-2) directly from 1/B Shubnikov-de Haas oscillation frequency B_F (Tesla).
Read the guide →Lookup & ReferenceShuffle Rows & Draw a Sample Without Replacement (SORTBY + TAKE)
Sort rows by frozen random keys and select K records without replacement. Validate sample size, retain unique record IDs, and distinguish repeated names from repeated records.
Read the guide →Engineering & ScienceSieder-Tate Laminar Tube Nusselt Number With Wall-Viscosity Correction
Calculate the Sieder-Tate developing laminar tube Nusselt number Nu_D = 1.86 * (Re * Pr * (D / L))^(1/3) * (mu_b / mu_w)^0.14 to capture viscous lube-oil heating enhancement vs. cold-wall boundary thickening.
Read the guide →Engineering & ScienceSieverts' Law: Diatomic Gas Solubility (H2, N2, O2) in Molten Metals & Vacuum Degassing
Calculate dissolved atomic hydrogen or nitrogen concentration [X] = K_S * SQRT(P_X2) in ppm for steelmaking, aluminum foundry degassing, and TIG/MIG welding, plus target vacuum degassing pressure.
Read the guide →MathSIGN function: classify variances into +1, 0, or -1 directional flags
Turn any numeric delta into -1, 0, or +1 with SIGN to map status labels via CHOOSE or count net gains vs declines.
Read the guide →Math & RoundingSign-Independent Directed Rounding Toward +Infinity and -Infinity (CEILING.PRECISE & FLOOR.PRECISE)
Round positive and negative numbers strictly toward +infinity or -infinity using ABS(significance) and INT identities that never fail with #NUM! when value and step have opposite signs.
Read the guide →OperationsSilver-Meal Heuristic: Dynamic MRP Lot-Sizing Cost Per Period
Choose the next production lot with the Silver–Meal first-increase rule, compare unrounded average costs, and calculate its quantity in a four-period worksheet.
Read the guide →Healthcare & ClinicalSilverman-Andersen Neonatal Respiratory Distress Score (0–10 NICU Formula)
Calculate William Silverman and Dorothy Andersen’s 0–10 newborn work-of-breathing retraction score across Upper Chest Movement, Lower Intercostal Retractions, Xiphoid Retraction, Nasal Flaring, and Expiratory Grunting to guide NICU bubble CPAP vs. intubation.
Read the guide →Biostatistics & Oncology TrialsSimon Two-Stage Phase II Futility Stopping and Expected Sample Size
Evaluate Simon’s (1989) Optimal vs Minimax single-arm Phase II designs via Stage-1 early futility probability PET(p_0) and expected sample size EN(p_0) = n_1 + (1 - PET(p_0)) * n_2.
Read the guide →FinancialSimple interest formula: I = P * r * t for days, months, and years
Calculate non-compounding interest and maturity payoff on promissory notes under Actual/365 and commercial Actual/360 conventions.
Read the guide →Healthcare & Clinical CalculationsSimplified PESI (sPESI) Pulmonary Embolism 30-Day Mortality & Outpatient Score
Calculate David Jiménez’s ESC-recommended 0–6 Simplified Pulmonary Embolism Severity Index (sPESI) across Age > 80, Cancer, Chronic Cardiopulmonary Disease, HR >= 110, SBP < 100, and SpO2 < 90% to identify Low-Risk PE (sPESI = 0) for direct DOAC outpatient treatment.
Read the guide →EngineeringSimply supported beam deflection (5wL^4/384EI), bending moment & L/360 check
Explore ideal simply supported beam deflection and bending moment, with an illustrative span/360 comparison rather than a structural approval.
Read the guide →StatisticalSimpson's Diversity Index (1 - D) & Lincoln-Petersen Mark-Recapture formulas
Calculate finite-sample diversity and a Chapman capture–recapture estimate under explicit sampling assumptions.
Read the guide →MathSIN function: calculate vertical rise from sloped length and angle
Convert degree angles with RADIANS before calling SIN to compute ramp elevation gain and wave amplitudes.
Read the guide →Polymer Processing, Plastics Extrusion & Rheology EngineeringSingle-Screw Extruder Drag Flow & Die Operating Point Formula
Calculate Open-Discharge Drag Flow Q_d (L/hr), Coupled Screw-Die Operating Throughput Q_op (L/hr), Back-Pressure Throttle Loss (%), and Die Head Melt Pressure Delta_P (MPa).
Read the guide →FinancialSinking Fund Payment (PMT) & Growing Perpetuity Valuation CF1 / (r - g) formulas
Calculate periodic sinking-fund savings needed to retire a future bond/capex obligation via PMT(r, n, 0, -FV) and value infinite cash flows via CF1 / (r - g).
Read the guide →Powder Metallurgy, Metal Injection Molding (MIM) & Technical Ceramics, Sintering ShrinkageSintering Linear Shrinkage & Die Tooling Oversize Formula
Calculate Volumetric Sintering Shrinkage (%), Isotropic Linear Shrinkage 1 - (rho_g/rho_s)^(1/3) (%), Tooling Oversize Scale Factor, and Green Die Cavity Dimension (mm).
Read the guide →Operations, Logistics & EngineeringSips (Langmuir-Freundlich) & Redlich-Peterson 3-Parameter Adsorption Isotherm Formula
Bridge heterogeneous Freundlich low-concentration behavior and finite Langmuir monolayer saturation using Robert Sips’s (1948) isotherm q_e = q_max*(K_S*C_e)^n / (1 + (K_S*C_e)^n) and the Redlich-Peterson (1959) hybrid equation.
Read the guide →Epidemiology & BiostatisticsSIR Herd Immunity Threshold & Vaccine Efficacy Formula
Calculate the Kermack-McKendrick Herd Immunity Threshold 1 - 1/R_0, Effective Reproduction Number R_eff = R_0*(1 - v*VE), and critical coverage v_crit = (1 - 1/R_0)/VE.
Read the guide →Clinical Medicine & PhysiologySiri (1956) & Brozek (1963) Body Density to Body Fat % Formula: Hydrostatic & BodPod
Convert two-compartment whole-body density D_b (g/cm^3) from hydrostatic underwater weighing, air-displacement plethysmography (BodPod), or skinfold equations into Percent Body Fat (%BF) and Fat-Free Mass using Siri (495/D_b - 450) and Brozek (457/D_b - 414.2).
Read the guide →Healthcare & ClinicalSIRS (Systemic Inflammatory Response Syndrome) 4-Criterion Score Formula
Evaluate Bone’s classic 0–4 SIRS criteria—Temperature (> 38.0°C or < 36.0°C), Heart Rate (> 90 bpm), Tachypnea/Hyperventilation (RR > 20/min or PaCO2 < 32 mmHg), and Leukocyte/Band abnormality (WBC > 12, < 4, or Bands > 10%)—with >= 2 SIRS-Positive screening.
Read the guide →StatisticalSix Sigma DPMO and Rolled Throughput Yield (RTY) formulas
Calculate Defects Per Million Opportunities (DPMO) across multi-checkpoint units and chain step-by-step First Pass Yields with PRODUCT.
Read the guide →Healthcare & Clinical FormulasSix-Item Screener (SIS) Verbal Cognitive & Delirium/Dementia Score
Calculate Christopher Callahan’s 1-minute verbal Six-Item Screener (0–6: Year, Month, Day of Week + 3-Item Delayed Recall) requiring zero pen/paper for telephone, ambulance, and ED cognitive screening.
Read the guide →Sports Analytics & Quantitative ModelingSkellam Goal Difference & Dixon-Coles Tau Formula
Calculate Skellam Poisson Goal-Difference Mean (lambda_H - lambda_A) and Variance (lambda_H + lambda_A), plus Dixon & Coles (1997) low-scoring correlation adjustment for 0-0 and 1-1 soccer draws.
Read the guide →Engineering & Fluid MechanicsSkempton’s Pore-Pressure Parameters B & A Formula: Undrained Triaxial Soil Saturation & Shear
Verify triaxial specimen saturation (B >= 0.95) and diagnose contractive vs. dilative undrained shear behavior using Alec W. Skempton’s 1954 equation delta_u = B * [delta_sigma3 + A * (delta_sigma1 - delta_sigma3)].
Read the guide →Clinical Medicine & PhysiologySlaughter Pediatric Skinfold Body Fat % Calculator: Triceps + Subscapular (Ages 8–18)
Estimate child and adolescent Percent Body Fat (%BF) from Triceps + Subscapular skinfolds S_2 (mm) using Mary H. Slaughter et al.’s (1988, Human Biology) multicomponent-validated quadratic (S_2 <= 35 mm) and linear (S_2 > 35 mm) youth equations.
Read the guide →Healthcare & Clinical CalculatorsSLEDAI-2K (Systemic Lupus Erythematosus Disease Activity Index 2000)
Compute weighted SLEDAI-2K lupus disease activity via SUMPRODUCT across 8-, 4-, 2-, and 1-point neurovascular, renal/articular, mucocutaneous/serologic, and constitutional descriptors.
Read the guide →FinancialSLN function: calculate annual straight-line depreciation after salvage value
Spread fixed-asset cost minus salvage value evenly across useful life periods with SLN.
Read the guide →Equity Valuation & Forensic AccountingSloan Accrual Anomaly & Cash-ROA Persistence Formula
Decompose Return on Assets into Cash-Flow ROA (CFO / Avg_Assets) and Operating Accrual Ratio ((NI - CFO) / Avg_Assets) to forecast next-year ROA under Richard Sloan’s (1996) differential persistence model.
Read the guide →StatisticalSLOPE function: estimate variable cost per unit or trend rate of change
Calculate ordinary least-squares linear slope (change in Y per unit of X) using SLOPE—without reversing the Y and X arguments.
Read the guide →Charts & Visual DataSlopegraph & Bump Chart Formula: Tie-Safe Ordinal Rank & Rank-Jump Delta
Stage multi-period Bump Charts and Tufte Slopegraphs by computing tie-broken descending ranks RANK + COUNTIF_prior - 1 and signed rank-climb deltas Rank_Prior - Rank_Current.
Read the guide →Biosolids Management, Decanter Centrifuges & Mass BalanceSludge Centrifuge Solids Capture % & Cake Tonnage Formula
Calculate Decanter Centrifuge / Belt Press Solids Capture Recovery (%) from Lab %TS Samples, Daily Dry Solids Captured (t/d), Hauled Wet Cake Tonnage (t/d), and Volume Reduction (%).
Read the guide →Wastewater Process Control, Activated Sludge & Secondary ClarificationSludge Volume Index (SVI) & Return-Sludge RAS Ratio Formula
Calculate Mohlman Sludge Volume Index SVI (mL/g), Theoretical Maximum Return-Sludge Concentration X_R (mg/L), Minimum RAS Recycle Ratio Q_r/Q, and Filamentous Bulking Status.
Read the guide →Healthcare & ClinicalSMART-COP Severe Pneumonia Score (ICU Ventilator & Vasopressor Risk)
Calculate the 0–11 point Australian SMART-COP community-acquired pneumonia score to predict Intensive Respiratory or Vasopressor Support (IRVS) need across Low, Moderate, High, and Very High tiers.
Read the guide →Operations & LogisticsSMED Setup Reduction Formula: Internal-to-External Changeover & Lot Shrinking
Quantify Shigeo Shingo’s Single-Minute Exchange of Die (SMED) conversion—separating external setup from stopped-machine internal downtime to shrink economic lot sizes by SQRT(T_new / T_old) or unlock shift availability.
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