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Chiang Life-Table q_x & Sullivan DFLE Formula
Convert observed age-group mortality rates n_m_x into exact interval death probabilities n_q_x = (n*n_m_x)/(1 + n*(1 - n_a_x)*n_m_x) via Chiang (1968) and compute Sullivan healthy person-years.
Read the guide →Engineering & ScienceChild-Langmuir Three-Halves (V^3/2) Space-Charge Current Limit & Perveance
Calculate space-charge-limited electron current density J = 0.002334 * (V^1.5) / (d^2) in mA/cm^2 and diode perveance P = I / V^1.5 for vacuum tubes, X-ray anodes, and ion thruster grids.
Read the guide →StatisticalChild-Pugh spreadsheet arithmetic with validated inputs and class boundaries
Practice nested IF with a conventional non-cholestatic Child-Pugh scoring example using fictional data and explicit units.
Read the guide →Engineering & ScienceChirikov Resonance-Overlap Criterion: Hamiltonian Chaos & Standard Map Parameter
Evaluate Boris Chirikov’s resonance-overlap parameter s = (Δω_1 + Δω_2) / Ω_12 and Chirikov-Taylor standard-map stochasticity threshold K_c ≈ 0.9716.
Read the guide →Electrochemical Engineering, Chlor-Alkali & Industrial ElectrolyzersChlor-Alkali Membrane Cell NaOH, Cl2 ECU & kWh/Ton Formula
Calculate Daily 100%-Basis Caustic Soda NaOH Production (t/d), Co-Product Chlorine Gas Cl2 (t/d at 0.886 ECU Ratio), Rectifier DC Load (MW), and Specific DC Energy (kWh/t NaOH).
Read the guide →StatisticalChoose arithmetic, geometric or harmonic means for the right weighting
Compute Geometric Mean PRODUCT(x)^(1/n) for compounded growth factors and Harmonic Mean COUNT(x)/SUM(1/x) for speeds, rates, and P/E multiples.
Read the guide →Logical decisionsCHOOSE function: switch forecast scenarios with a 1-based selector
Pick the 1st, 2nd, or 3rd assumption cell from a numbered scenario input, and catch out-of-range selector numbers that trigger #VALUE!.
Read the guide →Dynamic arraysCHOOSECOLS function: extract, reorder, and pull last columns dynamically
Select non-adjacent columns in any order or reference columns from the right with negative indexes using CHOOSECOLS.
Read the guide →Dynamic arraysCHOOSEROWS function: pull specific rows or the latest entry from the bottom
Extract milestone rows by position or fetch the last row of a growing log with CHOOSEROWS and negative row indexes.
Read the guide →Economic Geography & Urban SystemsChristaller K=3, 4, 7 & Beckmann City-Size Formula
Calculate Walter Christaller’s (1933) hexagonal Central Place market-area hierarchy K^(m - 1) and Martin Beckmann’s (1958) recursive urban system population T_m = (R_1 / (1 - k_srv)) * (K / (1 - k_srv))^(m - 1).
Read the guide →Engineering & ScienceChun-Seban Falling-Film Evaporator Nusselt Correlation (Wavy vs. Turbulent)
Calculate vertical falling-film evaporator dimensionless Nusselt number Nu_f across wavy-laminar (0.821 * Re_f^-0.22) and turbulent (0.0038 * Re_f^0.4 * Pr^0.65) regimes.
Read the guide →Engineering & ScienceChung-Lee-Starling Polar & Acentric Gas Viscosity Formula (F_c Shape Factor)
Calculate dilute gas viscosity η = 4.0785 * F_c * SQRT(M * T) / (V_c^(2/3) * Ω_v) (µPa·s) accounting for Pitzer acentricity ω, reduced dipole μ_r, and hydrogen-bonding κ!
Read the guide →EngineeringChurchill-Bernstein Cylinder Crossflow Nusselt Number Formula
Estimate average Nusselt number and external convection coefficient for a circular cylinder in crossflow, using validated inputs and unrounded intermediate results.
Read the guide →Engineering & ScienceChurchill-Chu Vertical Plate Natural Convection Formula (All Rayleigh Numbers)
Calculate free-convection heat transfer Nu_L = [0.825 + 0.387*Ra^(1/6) / (1 + (0.492/Pr)^(9/16))^(8/27)]^2 across laminar and turbulent buoyancy boundary layers.
Read the guide →Foundry Metallurgy, Green-Sand Casting Solidification & Shrinkage Riser DesignChvorinov Sand-Casting Modulus & Riser Sizing Formula
Calculate Casting Cooling Modulus M_c = V/A (cm), Chvorinov Solidification Time t_s = B*(V/A)^n (min), Minimum Blind Cylindrical Riser Diameter D_r = H_r (cm), and Riser Feed Volume (cm³).
Read the guide →Architectural Illumination & Ergonomics, CIE 117 / EN 12464-1 Unified Glare Rating (UGR)CIE Unified Glare Rating (UGR) Discomfort Formula
Calculate Single-Luminaire Guth Glare Term (L²*omega/p²), Cumulative Multi-Fixture Glare Sum, CIE 117 Unified Glare Rating UGR, and EN 12464-1 <=19 Office Compliance.
Read the guide →Econometrics & Omitted-Variable SensitivityCinelli-Hazlett Partial R-Squared and Robustness Value (RV) Formula
Convert any regression t-statistic and degrees of freedom into partial R^2_{Y~D|X} and Cinelli & Hazlett’s (2020) omitted-variable Robustness Value RV_q.
Read the guide →Engineering & ScienceCipolletti Trapezoidal Weir & Parshall Flume Free-Flow Discharge Formulas
Measure irrigation canal and wastewater plant flow rates (m³/s) with Cesare Cipolletti’s 1H:4V end-contraction-compensated weir Q = 1.859*L*H^1.5 and Ralph Parshall’s throat flume Q = C*H_a^n.
Read the guide →Healthcare & ClinicalCISNE Score Calculator: Stable Solid-Tumor Febrile Neutropenia (0–8)
Calculate the ASCO/IDSA-endorsed Clinical Index of Stable Febrile Neutropenia (CISNE, 0–8 points) to catch occult decompensation missed by MASCC in solid-tumor chemotherapy outpatients.
Read the guide →Environmental Engineering, Water/Wastewater Clarification & HydraulicsClarifier Surface Overflow Rate (SOR) & Weir Loading Formula
Calculate Circular Clarifier Settling Plan Area (ft²), Camp-Hazen Surface Overflow Rate SOR (GPD/ft²), Hydraulic Detention Time HRT (hrs), and Peripheral Weir Overflow Rate (GPD/ft).
Read the guide →Spatial Statistics & EcologyClark-Evans Nearest-Neighbor Dispersion Index (R) and Z-Score
Classify point patterns as Clustered (R < 1), Random Poisson CSR (R = 1), or Regular Lattice (up to R = 2.1491) using Clark and Evans’s (1954) nearest-neighbor formula.
Read the guide →Supply Chain Operations, Multi-Echelon Inventory & Risk PoolingClark-Scarf Echelon Stock & Square-Root Risk-Pooling Formula
Calculate Clark-Scarf Multi-Echelon System Inventory Position (DC_OnHand + In_Transit + Downstream_Store_Stock), Decentralized Store Safety Stock N*z*sigma, Echelon-Pooled Safety Stock SQRT(N)*z*sigma, and Centralization Buffer Savings.
Read the guide →Quantitative Finance & ForecastingClark-West Nested MSPE-Adjusted Test and Out-of-Sample R-Squared
Evaluate nested predictive regressions against a historical-mean or random-walk benchmark using Campbell-Thompson R^2_OS and Clark & West’s (2007) noise-adjusted f_t statistic.
Read the guide →Transportation Logistics & Operations ResearchClarke-Wright VRP Route Savings s_ij Formula
Calculate Geoff Clarke and John Wright’s (1964) pairwise route-consolidation distance savings s_ij = MAX(0, d_0i + d_0j - d_ij) gated by truck payload capacity Q_veh.
Read the guide →StatisticalClassify & Flag Scores by Quartile Tier (QUARTILE.INC & IQR)
Calculate inclusive quartile cutoffs and IQR, then classify scores into threshold bands with an explicit policy for ties. Bands do not necessarily contain equal numbers of observations.
Read the guide →Data Validation & HygieneClassify and Count Formula Error Codes (#N/A vs. #DIV/0! and #VALUE!) for Workbook QA
Separate benign missing-lookup #N/A flags from severe arithmetic or data-type faults (#DIV/0!, #VALUE!) using ISNA and ISERROR priority triage.
Read the guide →Date & TimeClassify elapsed calendar days into 0–30, 31–60, 61–90 and 91+ buckets
Calculate elapsed days between an onboarding, ticket, or trial start date and a report cutoff date, then map into clean tenure bands via table-driven XLOOKUP.
Read the guide →Refinery Gas Treating, Sulfur Recovery Units (SRU) & EmissionsClaus Sulfur Recovery Unit (SRU) 1/3 Air Split & Yield Formula
Calculate Exact One-Third H2S Combustion O2 Demand (kmol/hr), Reaction-Furnace Dry Air Rate (kmol/hr), Recovered Liquid Elemental Sulfur (t/d), and Tail-Gas Incinerator SO2 (t/d).
Read the guide →EngineeringClausius-Clapeyron Vapor Pressure & Raoult's Law binary mixture formulas
Predict pure-component vapor pressure across temperature via P2 = P1 * EXP((-dHvap/R) * (1/T2 - 1/T1)) and ideal mixture bubble pressure via Raoult's Law.
Read the guide →Engineering & ScienceClay Primary Consolidation Settlement (Sc) & Terzaghi Time Factor (Tv)
Calculate normally consolidated clay ultimate settlement Sc = (Cc * H0 / (1 + e0)) * LOG10((sigma0 + dsigma) / sigma0) in mm and Terzaghi dimensionless time factor Tv = (cv * t) / Hdr^2.
Read the guide →TextClean and format 10-digit phone numbers with formulas
Strip spaces, parentheses, hyphens, and dots from imported phone strings, validate 10-digit length, and format uniformly.
Read the guide →TextCLEAN function: strip line breaks and control characters without fusing words
Remove non-printable ASCII 0-31 characters such as CHAR(10) line feeds with CLEAN, and avoid accidentally joining words across wrapped lines.
Read the guide →Wholesale Electricity Trading & Carbon MarketsClean Spark Spread, Clean Dark Spread & Fuel-Switch Formula
Calculate Wholesale Power Gross Spark Spread, Carbon-Adjusted Clean Spark Spread (Gas CCGT) vs Clean Dark Spread (Coal), and the Breakeven Carbon Allowance Fuel-Switching Price.
Read the guide →Healthcare & Clinical CalculatorsCLIF-C ACLF Score (Acute-on-Chronic Liver Failure 28-Day Mortality)
Calculate Jalan et al.’s (2014) EASL-CLIF Consortium ACLF prognostic score = 10 * (0.33 * CLIF-OF + 0.04 * Age + 0.63 * LN(WBC_10^9_L) - 2) from CLIF-Organ Failure subscore (6–18), age, and leukocyte count.
Read the guide →Healthcare & Clinical CalculatorsCLIF-C AD Score: Acute Decompensation of Cirrhosis (Non-ACLF 90-Day Risk)
Calculate Rajiv Jalan et al.’s (2015) EASL-CLIF Acute Decompensation score = 10 * (0.03*Age + 0.66*LN(Cr) + 1.71*LN(INR) + 0.88*LN(WBC) - 0.05*Na + 8) for hospitalized cirrhotic patients without ACLF.
Read the guide →Statistical & PsychometricsClinical Trial Fragility Quotient (FQ) & Attrition-vs-Fragility Audit
Standardize Michael Walsh et al.’s (2014) RCT Fragility Index across trial sample size via Fragility Quotient FQ (%) = (FI / N) * 100 and audit whether Lost-to-Follow-Up exceeds FI!
Read the guide →Structured Finance, CLO Securitizations & Cashflow Waterfall CovenantsCLO Overcollateralization (OC), IC Test & Cure Paydown Formula
Calculate Collateralized Loan Obligation (CLO) Tranche Overcollateralization (OC) Ratio, Par Cushion Over Trigger ($M), Interest Coverage (IC) Ratio, and Senior Note Paydown Cure Required via Equity Interest Diversion.
Read the guide →Business & OperationsClosed Queueing Network Throughput: Zahorjan-Lazowska Balanced Job Bounds (BJB)
Bracket exact Mean-Value-Analysis / Buzen closed-network throughput X(N) for fixed-population pallet loops and database thread pools between tight lower bound N / (D_tot + (N - 1) * D_max) and upper bound MIN(1 / D_max, N / (D_tot + (N - 1) * D_avg)).
Read the guide →Closed-Die Impression Forging, Automotive Crankshaft/Connecting-Rod Tooling & Flash DesignClosed-Die Forging Flash-Land Factor & Tonnage Formula
Calculate Flash-Land Width-to-Thickness Aspect Ratio w/t, Cavity Multiplying Factor Kf = 3.0 + 1.15*(w/t), Mean Cavity Die Pressure (MPa), and Required Mechanical Press Capacity (metric tons).
Read the guide →Asset Management, Income Investing & 1940-Act FundsClosed-End Fund NAV Discount & Market Yield Formula
Calculate Closed-End Fund (CEF) Share Price Discount/Premium to NAV (Price - NAV)/NAV, Required Portfolio NAV Distribution Rate vs Amplified Market Distribution Yield, and 1940-Act Structural Leverage Ratio.
Read the guide →Commercial Mortgage & Fixed-Income AnalyticsCMBS Yield Maintenance Prepayment Penalty Formula
Calculate commercial mortgage Yield Maintenance prepayment penalties MAX(Floor * Balance, Balance * (r_note - y_T) * ((1 - (1 + y_T)^(-T)) / y_T)) as Treasury yields move.
Read the guide →Structural Engineering, TMS 402 Reinforced CMU Masonry Walls & Buckling SlendernessCMU Masonry Wall Prism Strength & Slenderness Formula
Calculate Masonry Wall Slenderness Ratio h/r, TMS 402 Piecewise Parabolic-to-Euler Buckling Reduction Factor, Nominal Axial Capacity Pn (kN), and LRFD Strength phi*Pn (kN).
Read the guide →5-Axis CNC Die/Mold Machining & CAM Toolpaths, Ball-Nose Scallop & Effective DiameterCNC Ball-Nose Scallop Cusp Height & Effective Vc Formula
Calculate 3D Mold Finishing Scallop Cusp Height h (um), Effective Engaged Cutting Diameter Deff (mm), True Cutting Surface Speed Vc,eff (m/min), and % of Nominal Diameter Speed.
Read the guide →Mechatronics, CNC Linear Motion Axes, Precision Ball Screws & Servo SizingCNC Ball-Screw Lead, RPM & Servo Motor Torque Formula
Calculate Peak Rapid-Traverse Ball-Screw Speed (RPM), External Thrust Load Torque T_load (N·m), Double-Nut Preload Drag Torque T_pre (N·m), and Continuous Servo Motor Torque (N·m).
Read the guide →CNC Machining, High-Efficiency Milling (HEM), Trochoidal Toolpaths & CAM FeedsCNC Milling Radial Chip-Thinning Factor & MRR Formula
Calculate Spindle Speed N (RPM), Exact Radial Chip-Thinning Compensation Factor RCTF, Compensated Table Feed Rate V_f (mm/min), and Volumetric Material Removal Rate MRR (cm³/min).
Read the guide →Industrial Organization & Pricing StrategyCoase Durable-Good Monopoly: Leasing vs Spot Sale Formula
Quantify Ronald Coase’s (1972) durable-good time-inconsistency profit loss by comparing 2-period commitment leasing (1 + delta)*(a^2 / 4) against spot selling a^2 * (2 + delta)^2 / (4 * (4 + delta)).
Read the guide →FinancialCobb-Douglas production function (Y = A * K^alpha * L^beta), MPK & MPL
Calculate economic output Y, Marginal Product of Capital (MPK), Marginal Product of Labor (MPL), and Returns to Scale from TFP, Capital, and Labor.
Read the guide →Operations Research & Queueing TheoryCobham Non-Preemptive Priority Queue Waiting Formula
Calculate urgent W_q1 = R_0/(1 - rho_1) and routine W_q2 = R_0/((1 - rho_1)*(1 - rho_1 - rho_2)) M/G/1 waits via Alan Cobham’s (1954) priority formula.
Read the guide →Statistical & EconometricCochran-Armitage Trend Test (Z_CA & χ²_trend) for Dose-Response Proportions
Test monotonic dose-response trends across ordered 2×k clinical trial cohorts using the Cochran-Armitage statistic Z_CA = T / SQRT(Var(T)) and 1-df χ²_trend = Z_CA^2.
Read the guide →Nonparametric Statistics & Clinical TrialsCochran’s Q Test Formula for Matched Binary Outcomes Across k Conditions
Test whether binary pass/fail proportions differ across three or more matched treatments or raters using Cochran’s (1950) Q statistic and verify unanimous-row invariance.
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