Calculator User Manual

1 · Quick Start

The tool has twelve workspaces along the top. Guided Analysis opens first and walks you from a dataset to an analysis and next steps:

• Guided Analysis — import Excel/CSV or paste a table, reuse datasets, select a question, inspect assumptions and exclusions, and download a reproducible report. Start with a worked example and select Run guided analysis.
• Scientific Calculator — a full natural-display calculator that operates like a Casio fx-300ES PLUS 2: you type into stacked fractions, radicals, and exponents, with a SHIFT/ALPHA keypad, memory registers, and a history tape.
• Distributions — probability calculator for 16 distributions with live shaded-region plots, critical values, and reliability metrics.
• Distribution Lookup Tables — browse the exam reference tables, enter lookup keys and highlight the matching row, column and cell. See section 25 for tail conventions and examples.
• Hypothesis Tests — ten test presets that compute the statistic, p-value, and decision, and draw the rejection region.
• Data & Quality — raw-data summaries, regression, capability, I-MR, ANOVA and chi-square.
• Advanced SPC — choose a chart for your sampling design, establish a baseline, monitor later observations and review capability intervals where supported. See section 23.
• Cost & Savings — review COQ/COPQ, compare costs at a common production mix, evaluate cash scenarios and track documented benefits. See section 24.
• Graphing — function, parametric, polar and sequence plots, tables and numerical calculus.
• Lists & Matrices — advanced expressions, complex numbers, units and a supported-function catalog.
• Finance — TVM and amortization, with NPV/IRR in the math workspace.
• Programs — bounded calculation programs using the documented command subset.

2 · The Display

The screen has four zones, top to bottom:

3 · Typing Equations — the Natural Display Editor

The equation area is a structural editor: the math on screen is the input itself, not a preview. You can drive it entirely from the keypad, entirely from your keyboard, or mix both.

Creating structures

Moving around

• ◀ ▶ walk through the structure — into a radical, across the radicand, out the other side.
• ▲ ▼ hop between numerator and denominator, and between a root's index and radicand. ▼ also exits an exponent.
• Click anywhere — on a character (cursor lands before/after depending on which half you click), on an empty dashed slot, or on a slot's padding — the cursor goes exactly there.
• Backspace / DEL deletes leftward; at the edge of a structure it steps inside; deleting a fraction bar dissolves the structure but keeps its contents.
• Ctrl+Z undoes (up to 80 steps). Ctrl+C with nothing selected copies the linear form. Pasting a linear expression like sqrt(10.39/(0.25*tan(720/(2*6))*6)) converts it into the stack automatically.
• Enter = evaluate.

Keyboard / builds a stacked fraction; the keypad ÷ inserts a linear ÷ sign — exactly the distinction the real Casio makes between the fraction key and the divide key. Both compute identically.

4 · The Keypad & SHIFT / ALPHA System

The keypad mirrors the fx-300ES PLUS 2 faceplate. Every key can carry up to three functions:

• Main label (colour follows the selected theme) — what the key does when pressed alone.
• Gold legend above-left — the SHIFT function.
• Red legend above-right — the ALPHA function (mostly the A–F registers).

SHIFT and ALPHA are one-shot prefix keys: press one (its badge lights and the key outlines), press the target key, and the modifier clears itself. Pressing it again cancels. hyp works the same way as a prefix for the trig keys: hyp → sin gives sinh, and hyp → SHIFT → tan gives tanh⁻¹.

The circular replay D-pad navigates the equation (◀▶ through the structure, ▲▼ between stacked levels). On an empty screen, ▲ recalls your previous calculation for editing — the classic Casio replay.

5 · Complete Key Reference

6 · Memory & Registers

Seven user storage registers: the running memory M and six registers A–F, plus the automatic previous-answer value Ans.

You can also just type a register: uppercase A–F or M from the keyboard inserts it (lowercase e is always Euler's constant; uppercase E is the register).

7 · Behavior After Pressing =

The Scientific Calculator follows these entry rules:

• Digit or function → starts a fresh calculation.
• Operator (+ − × ÷ ^ % !) → continues from Ans: after 2+2=, pressing +3= gives 7.
• Fraction key → the answer becomes the numerator.
• ◀ or click → edit the expression you just evaluated.
• ▲ on an empty screen → replay: recalls the last entry.

8 · Display Modes

9 · History Tape & STAT Data Panel

History tape (right panel): every evaluated calculation is kept, rendered as a mini stacked equation with its result, and persists between sessions on the website. Click any entry to reload it into the editor for modification. Ans always holds the most recent result.

STAT panel (below the tape, or SHIFT+1): paste data — one value per line or comma-separated — and press Compute stats for n, x̄, s (sample), σ (population), Σx, Σx², min / Q1 / median / Q3 / max. Select X,Y pairs explicitly and paste at least three pairs (one pair per line) for linear regression: slope b, intercept a, r, and r² — the calculator's equivalent of the Casio STAT mode, with paste instead of one-at-a-time entry.

10 · Distributions Tab

Pick a distribution, set its parameters, and ask either question:

• Given x → probability — enter a value; get the PDF/PMF, left tail P(X ≤ x), right tail P(X ≥ x), the equal-tail two-tailed probability, or P(a ≤ X ≤ b) in Between mode.
• Given probability → x — enter α or a central probability; get critical values (z₀.₀₂₅ = 1.959964, t-table values, χ² and F criticals) or a central interval.

The plot shades the selected region in amber, labels the boundary values on the axis, prints the shaded probability, and marks the mean with a dashed teal line. Discrete distributions (Binomial, Poisson, Geometric, Negative Binomial, Hypergeometric) draw as bars with the included bars shaded — and the right tail is computed inclusively, P(X ≥ k), as it should be for counts.

Discrete x inputs must be integers; distribution count parameters must also be integers and satisfy their domain. Uniform bounds require lower < upper. Two-tail mode is disabled for discrete distributions because equal tail regions cannot generally match the requested probability. In inverse discrete mode, the selected quantile is the smallest integer with CDF at least the target; the actual inclusive region probability is displayed separately and may differ from the requested value.

Reliability distributions

For Exponential, Weibull, Lognormal, and Gamma, the results panel adds R(t) survival probability, hazard rate h(t), and MTBF. Weibull results include the β interpretation (β < 1 infant mortality, β = 1 random failures, β > 1 wear-out) plus B10 life, and a reminder that F(η) = 63.2% always.

11 · Hypothesis Tests Tab

Ten presets: 1- and 2-sample z (known population standard deviations), 1-sample t, 2-sample t (Welch and pooled), paired t, 1- and 2-proportion z, χ² for one variance, and F for two variances. Enter the summary statistics, choose the alternative (≠, <, >) and α, and run. You get the test statistic, p-value, critical value(s), and a plain-language decision — plus a plot of the reference distribution with the rejection region shaded and your statistic drawn as a red line. If the red line lands in the amber region, the result is significant at your α.

All fields are checked before calculation. Sample sizes and success counts must be whole numbers, standard deviations positive, and success counts no larger than their sample sizes. Changing a field, test or alternative clears previous results. Two-sided (1 − α) intervals accompany the tests, even when the selected hypothesis test is one-sided. Proportion intervals use the Wald approximation and can perform poorly near 0 or 1; sparse expected counts trigger a normal-approximation caution.

For the asymmetric χ² and F distributions, two-tailed p-values use the equal-tail-probability convention, matching Minitab.

12 · Keyboard Shortcut Summary

13 · Accuracy Notes

The legacy distribution engine uses incomplete gamma/beta functions and numerical inversion. Tiny upper tails are evaluated directly where supported. New numerical and interface tests cover known matrix, probability, calculus, finance, control-chart, regression and ANOVA cases. This is targeted validation, not a guarantee of uniform precision across every input. IEEE double precision and displayed rounding can affect extreme cases; a numerical failure or undefined statistic must not be treated as evidence about a process.

14 · Data & Quality

Choose an analysis before pasting data. Load example supplies a complete working dataset. Analyze data replaces the results; changing data or settings invalidates the previous results and disables exports.

Numbers may be separated by tabs, commas, semicolons or spaces. Use a period for the decimal point and omit headers, units, thousands separators and totals. Scientific notation such as 1.2e-3 is accepted. Blank lines are ignored, but empty cells and malformed values are rejected with their row and cell number. Data entry is limited to 10,000 values, 250,000 characters and magnitude 10¹⁰⁰.

6.9,5.4,5.8,4.6,4.0
8.3,6.8,7.8,9.2,6.5
8.0,10.5,8.1,6.9,9.3

Assumptions: mean intervals and regression inference require independent observations with suitable distributional assumptions. ANOVA assumes approximately normal errors and common population variance. Chi-square tests require independent counts in mutually exclusive categories. A large p-value does not prove equality or independence.

15 · Capability & I-MR Charts

1. Choose Capability & I-MR, then paste individual measurements in time order.
2. Enter the lower and/or upper engineering specification. A two-sided specification requires LSL < USL.
3. Review the I-MR plots and flagged observation numbers before interpreting capability.
4. Compare observed out-of-specification counts with the separate normal-model estimate.

Only points beyond the limits are flagged. Other run rules are not tested. No flags does not prove stability. The same data estimate the limits and populate the chart. Autocorrelation, mixed populations, small samples and non-normality can invalidate capability interpretations. Specification limits are engineering requirements; control limits describe process behavior. Values exactly on specification limits count as conforming.

Normal-model PPM uses the sample mean and overall s without a 1.5-sigma shift. It is not the observed defect rate. Zero variation is rejected; a very small sample cannot justify a confident capability claim.

49.6,47.6,49.9,51.3,47.8,51.2,52.6,52.4,53.6,52.1

16 · Graphing, Tracing & Tables

Press Draw graph. Zoom buttons rescale both axes around the window center. Standard window restores −10 to 10. Fit Y uses the central 96% of sampled Y values with padding, so extreme points can lie outside the fitted window. Trace a selected curve with the slider or arrow keys; click the plot to select the nearest sampled point on the selected curve. The trace reports sampled coordinates, not an exact interpolation.

Function tables support an explicit start, step and 1–200 rows. Undefined values are marked. Tables and numerical analysis are available in Cartesian function mode; parametric/polar/sequence results can be inspected through tracing. Download graph saves a standalone SVG.

Expressions accept pi, powers with ^, explicit multiplication with *, standard functions, and piecewise conditions such as x < 0 ? -x : x. Natural logarithm is log(x) or ln(x); base ten is log10(x). Expressions are sampled at 801 points. The drawing breaks across undefined samples and large jumps, but can still miss a narrow feature or discontinuity. It is not a symbolic proof.

Numerical analysis

17 · Lists, Matrices, Complex Numbers & Units

This workspace uses a locally served copy of math.js 15.2.0. It runs in a separate worker. Use the searchable supported-function catalog and the worked-example menu. Values may be assigned to named session variables; these variables are shared with graphing and programs. They are separate from the Scientific Calculator’s M/A–F registers and Ans.

RAD/DEG applies to sin/cos/tan, sec/csc/cot, their inverse functions, atan2, and the string-based nDeriv/fnInt/solve/seq helpers. Complex arg/angle always returns radians. Derivatives and integrals use the selected angle unit for the independent variable; for example, nDeriv("sin(x)",0) in DEG is π/180, and fnInt("sin(x)",0,180) is 360/π. Scientific Calculator angle mode is separate. Use parentheses and explicit operators when porting an expression from another calculator.

Object access, matrix/list indexing and user-defined function declarations are disabled. Whole-array operations, assignments and supported functions remain available. Arrays are limited to 10,000 cells; calculations time out after six seconds, resetting shared variables. Input is limited to 8,000 characters per evaluation. A calculation tape can be downloaded as text. Ans contains the last result, including the final result of a multi-line evaluation. Editing graph/table/program inputs clears the corresponding old output. Evaluating math expressions, running programs or clearing/resetting shared variables invalidates existing graphs so they can be redrawn from the current values; the calculation tape intentionally retains earlier calculations.

18 · Finance & Amortization

1. Select the unknown: N, annual nominal I%, PV, PMT or FV. Its input is disabled and ignored.
2. Enter the known values, payments per year, compounds per year, and beginning/end payment timing.
3. Use positive amounts for cash received and negative amounts for cash paid.
4. Press Solve TVM. For a positive principal and negative payment with an integer N ≤ 600, an amortization table is also displayed.

N counts payments, not years. If r is the effective rate per payment period, r = (1 + I% ÷ (100 × C/Y))^(C/Y ÷ P/Y) − 1. The solver uses PV(1+r)ᴺ + PMT × annuity factor + FV = 0, with the payment factor multiplied by (1+r) for beginning-of-period payments. At zero interest, the annuity factor is N.

In Lists & Matrices, npv(0.1,[-1000,400,400,400]) gives −5.25920360631. The first cash flow is at time zero; the rate is a decimal per period. irr([-1000,400,400,400],0,1) finds a periodic IRR of about 0.0970102574 within the bracket [0,1]. IRR may have multiple solutions or none. The TVM rate search starts just above the compounding-domain boundary of −100 × C/Y percent (with a small numerical margin) and expands its positive upper bound up to 40 times. It requires a sign-changing bracket and can fail for non-unique solutions or numerically extreme inputs. N is bracketed from 0 to 100,000; valid solutions outside that range are not found. Unbracketed or non-unique all-zero cash flows are rejected. PV/PMT/FV are solved directly; bracketed rate/N searches use scaled cash flows to avoid overflow.

eff(6,12) returns the annual effective percentage for 6% nominal compounded monthly. nom(effectivePercent,compoundsPerYear) reverses the conversion. Amortization uses unrounded internal cash flows; lender rounding, fees and payment conventions can differ.

19 · Calculation Programs

The program editor supports a deliberately limited, documented command language. It is not a complete TI-BASIC interpreter and does not load tokenized calculator files. Programs share session variables with the math workspace. Download source preserves the text for later reuse.

0→S
For(I,1,10)
  S+I→S
End
Disp S

This example prints 55. Supported commands are assignments with →, -> or =; Disp expression; ClrHome; For; While; If / Then / Else / End; and Stop. Every control block needs End. Conditions support comparisons and Boolean operators; a single = in an If/While condition means equality. Maximum: 500 lines, 10,000 execution steps and 1,000 output lines, plus the worker time limit.

Not supported: Goto/Lbl, interactive Input/Prompt/Menu, key scanning, drawing commands, list indexing, strings with TI-specific token semantics, assembly, hardware calls, .8xp imports, or installed TI applications. Unsupported statements produce errors; they are not silently skipped.

20 · TI-84 Plus CE Capability Map

The additions implement many comparable browser workflows. Full TI-84 Plus CE feature parity has not been reached. This is an independent UpSkill Sprint tool with its own interface, syntax and numerical behavior, not an emulator or a Texas Instruments product. Do not assume that a TI command or file will work without checking the supported catalog and this manual.

References: Texas Instruments guidebooks and TI commands reference. Exact TI behavior requires a compatible TI product; this tool makes no high-stakes-exam approval claim.

21 · Exports, Privacy & Troubleshooting

Data & Quality exports a text report with inputs, settings, metrics, tables and assumptions, plus a CSV results file. Guided Analysis additionally saves a workspace JSON containing its datasets, edits, exclusions and active analysis settings; it exports a traceable text report and CSV data/results. The graph exports SVG; the math tape and program source export text. Calculations run locally; these workspaces do not upload datasets or program source. The Scientific Calculator’s history continues to use browser local storage. Other session variables and analysis inputs are not automatically saved. Download anything you want to retain before reloading. Workspace files and reports contain the data you included; treat them accordingly when sharing.

• Invalid data: use the reported row/cell; remove headers and units, correct empty cells, and select the intended analysis.
• Old result disappeared: the input changed. Recalculate to create results for the current inputs.
• Worker timeout: reduce the expression/program size. The worker resets and shared variables must be re-entered.
• Graph gaps: inspect domain restrictions and singularities. Increase the window or evaluate the function numerically.
• Unsupported command: search the supported catalog. TI syntax is not automatically translated.
• Downloads: allow normal file downloads for this site. Export buttons for analysis remain disabled until calculation succeeds.

Manual links in the calculator open a new tab so you can keep your current inputs while reading. Closing or reloading the calculator still requires saving any workspace you want to retain.

Keyboard navigation: use Tab to reach tabs, then Left/Right or Home/End to switch workspaces. Range sliders support arrow keys. Result tables scroll horizontally on small screens; desktop tab bars wrap; on small screens, use the labelled workspace selector. Text and controls follow the site’s light/dark theme.

Method references

• NIST — Individuals control charts
• NIST — Process capability
• NIST — One-way ANOVA calculations (the example is independently calculated from the listed observations).
• Penn State — Chi-square independence
• math.js — expression syntax and expression security and worker isolation

Numerical results use finite-precision arithmetic. Very small positive probabilities are shown in scientific notation. An infinite endpoint density can be mathematically valid; it is not a probability mass. A failed convergence check is reported rather than used to make a hypothesis-test decision.

Numerical-range errors: very small variation, extreme fitted coefficients or ratios can exceed floating-point resolution even when individual inputs are finite. Rescale measurements to practical units (for example, metres to millimetres), update targets/specifications in the same units, and recalculate. Do not round nonzero variation to zero. Pearson correlation and Welch inference are checked for invariance under common positive unit changes. Goodness-of-fit observed counts must total at most 9,007,199,254,740,991; do not divide event counts merely to bypass this limit. Review the study or use software with suitable numerical precision.

22 · Guided Analysis Workspace

Use this workspace when you know the process question but want help selecting and interpreting an analysis. It reuses your data across questions and can send the included values to Data & Quality or Hypothesis Tests. Guidance is based on the study design you specify; assumptions cannot be certified from the data alone.

Bring and review your data

1. Paste a table from Excel, or select a CSV, TSV, text or .xlsx file. For Excel, choose a worksheet and press Add selected worksheet. Worksheet selection does not replace other datasets.
2. Set First row contains column names to match the file. Without headers, columns are named Column 1, Column 2, and so on. CSV supports quoted delimiters, escaped quotes and multiline cells. Choose the separator explicitly if automatic detection does not match your file.
3. Review the row count, blank cells and duplicates. Open Review and edit data to inspect cells, rename columns or correct values. The table displays 25 rows per page. Duplicate records are retained because repeated measurements can be legitimate.
4. Uncheck a row to explicitly exclude it. Row numbers and original source-row references are retained in the analysis report. Empty required cells block analysis unless you explicitly choose Use complete rows only; paired observations are always excluded as a pair. Invalid numeric text is never silently discarded.
5. Use Active dataset to switch among up to five datasets. Changes to data, column names, exclusions or analysis settings clear previous results and disable result exports until you rerun.

Limits: 10 MB per file; 10,000 data rows, 40 columns, 100,000 cells and 5 million text characters per dataset; 10,000 characters per cell. Excel rows can occupy rows 1–10,001 including a header. Import a clean rectangular table beginning at A1. Short CSV rows are padded with blank cells and reported; internal blank rows remain visible. Duplicate or blank column names are made unique with an import note. Decimal point: period; no thousands separators or embedded unit suffixes in numeric cells. Enter units separately in the analysis settings.

Excel values: the reader supports unencrypted .xlsx files with stored or deflated ZIP parts, subject to a 30 MB expanded-package limit. Older .xls, .xlsb, .xlsm, password-protected files and arbitrary spreadsheet features are not supported. Formula results are read from saved caches and may be stale; recalculate and save in Excel before importing. Missing formula caches are marked #FORMULA_WITHOUT_CACHED_VALUE. Formulas and external links are never executed. Hidden and filtered rows are included. Merged cells are not filled down. Percentage values use their stored decimal values (5% becomes 0.05). Common date/time formats become ISO text without an inferred timezone; duration formats and complex custom formatting require review. If the browser cannot decompress the workbook, paste the table or export CSV.

Choose the question and columns

The filter keeps one exact category in a selected column. Analysis grouping trims surrounding whitespace in category labels, while filtering uses the original exact value. Numeric-looking columns can still be selected as categories. For two independent groups, select a reference group: the reported difference is comparison minus reference. For matched pairs it is always after minus before.

Confidence choices are 90%, 95% and 99%, with significance levels 0.10, 0.05 and 0.01 respectively. All guided t tests are two-sided. Specify whether a higher or lower mean is desirable and a minimum useful change in the response units before examining the results. The interpretation reports whether the entire two-sided interval supports a favorable change beyond that threshold. It does not turn a nonsignificant result into evidence of equality or equivalence. The practical threshold does not change the hypothesis being tested, and is not used to interpret an overall ANOVA result.

Worked example — did the process change reduce thickness?

1. Load Process change — independent groups. The fictitious dataset contains six Before and six After observations.
2. The guide selects Thickness_mm as the response, Group as the grouping column and Before as the reference. Desired direction is lower; the practical threshold is 0.2 mm.
3. Review the independence assumptions and select Run guided analysis. Before mean = 10.55 mm; After mean ≈ 10.0167 mm; difference ≈ −0.5333 mm. The 95% Welch interval is approximately −0.7515 to −0.3152 mm.
4. The whole interval supports a reduction beyond 0.2 mm in this example. That conclusion still depends on a suitable study design and measurement system; before/after data alone do not prove causation.
5. Download the report or open the same included measurements in Hypothesis Tests. Return to Guided Analysis for the original labels, exclusions and interpretation.

Other examples cover matched measurements, time-ordered thickness and inspection outcomes by shift. They are fictitious teaching datasets, not process benchmarks.

Save, restore, export and reuse

• Save workspace downloads a versioned JSON file with all session datasets, import notes, source rows, edits, manual exclusions and the active question settings. No automatic browser or cloud saving occurs. Restore appends the saved datasets if the five-dataset limit allows; it never runs a saved analysis automatically. Rerun after review.
• Download data CSV includes every data row with Included and Source row columns. Retain the JSON project for exact restoration of settings and exclusions. CSV alone is a portable data table.
• Download analysis report contains the dataset revision, current settings, included source data, metrics, complete result tables, assumptions, interpretation, next steps and every excluded row with its reason.
• Download results CSV contains result metrics, complete tables, interpretation and exclusions. Text resembling a spreadsheet formula is prefixed with an apostrophe; legitimate numeric negatives remain numbers.
• Open in analysis tool copies current included values and matching confidence settings into Data & Quality or Hypothesis Tests. This is a snapshot: subsequent edits in either workspace do not update the other. Data & Quality retains its 10,000-numeric-value and 250,000-character input limits; the handoff is disabled when those limits would be exceeded.
• Long result tables show the first 200 rows on screen, and plots may show the first 1,500 points. Calculations and downloaded result tables use every included observation.

This workspace adds guided use of supported methods. For advanced SPC and capability intervals, see §23. Cost analysis and user-recorded benefit verification are in §24. It does not add measurement system analysis, mixed models for clustered data, power analysis, DOE or an AI interpretation service. For repeated observations from the same heat, coil or other experimental unit, select a method that accounts for the design before making inferential decisions.

Method references: NIST — two-sample t tests and Welch–Satterthwaite degrees of freedom; the existing capability, ANOVA and chi-square references in §21 also apply. The guided t-test reference fixtures were generated independently using SciPy 1.17.0.

23 · Advanced SPC & Process Capability

Open Advanced SPC to establish a historical baseline and evaluate later observations against its estimates. Choose a chart from the sampling design before looking for signals. These analyses operate locally in browser memory; no automatic cloud upload or project saving occurs.

Select the right chart and input

Paste numeric rows without headers; use commas, tabs, semicolons or spaces between columns and a period for decimals. Blank lines are ignored, but blank cells, nonnumeric tokens, negative counts and inconsistent row widths are rejected. Limits are 10,000 numeric values and 250,000 characters. p/np denominators must be positive integers, counts cannot exceed inspected units, and denominators/exposures are bounded at one billion.

Reuse Excel/CSV data

Import and clean the file in Guided Analysis. In Advanced SPC, expand Reuse data from Guided Analysis, show the active dataset, and enter column numbers in the required order. For example, 2,3,4,5,6 copies five measurement columns as a subgroup; 3,4 can copy nonconforming counts and inspected totals for p. The active dataset's explicit exclusions and exact category filter apply. Missing selected cells are rejected even if the guide allows complete-row inference: omitting time points changes a monitoring sequence.

The copy preserves source row references and records the dataset revision, columns, filter and omitted-row count. Subsequent edits in either workspace do not update the other. Editing the SPC text removes the old row mapping and labels the source as edited. Check order and subgroup meaning after any filtering; no sorting or regrouping is automatic.

Baseline review, freezing and monitoring

1. Select a chart and load its fictitious example or paste your process data. Confirm the sampling assumptions. Set the number of initial rows used as the baseline; only those rows estimate the center and variation.
2. Run Analyze process. Initial rows are labelled Baseline; remaining rows are Monitoring. Inspect both location and spread panels where provided, the phase boundary, and the point table. Small baselines carry a preliminary-limits warning.
3. Investigate baseline signals before approving limits. Freezing is disabled while the selected chart/rules flag baseline points. No point is automatically deleted and a signal does not establish its cause.
4. Select Freeze reviewed baseline. The baseline stays in a snapshot, the editor retains only later monitoring rows, and the chart type is locked. New observations cannot refit the center or spread. Save baseline JSON before closing the browser.
5. Paste the complete monitoring sequence when adding data. Every analysis restarts sequential chart state; it is not a live streaming service. EWMA starts at the baseline mean, CUSUM starts at zero, and moving ranges do not cross the baseline/monitoring boundary. CUSUM does not automatically reset after a signal.
6. Release baseline prepends its rows back to the editor and unlocks estimation. Document why a new baseline is justified instead of widening limits to absorb a process change.

Baseline JSON contains the original baseline values, chart type, source and available source-row labels. Restore recomputes estimates from the values rather than trusting saved limits. It does not restore monitoring data, specifications, confidence level, EWMA/CUSUM tuning or rule selections; review these controls separately. Restoring requests confirmation before clearing existing SPC input and never runs an analysis automatically. Keep a full SPC report for the exact analyzed settings and monitoring observations.

Limits and signal rules

Individuals use MR-bar/1.128 to estimate within variation. X-bar/R uses R-bar/d₂; X-bar/S uses S-bar/c₄. The paired spread chart has a zero-truncated lower limit. Attribute centers use total counts divided by total inspected units or exposure; p/u limits change with each row's denominator. These are symmetric three-sigma limits, truncated to the possible range; exact count-tail probabilities and dispersion corrections are not implemented.

EWMA uses λ (default 0.2) and L (default 3), with startup limits that approach their steady-state width. CUSUM uses standardized deviations with k = 0.5 and h = 5 by default; upper and lower accumulations are shown as positive magnitudes. These tuning choices are editable and change sensitivity; they are not calibrated average-run-length guarantees for an estimated baseline.

EWMA startup limits use a cancellation-resistant calculation for small λ. If the limits cannot be resolved numerically, analysis stops instead of publishing a zero-width band or false signal. Review λ and the measurement scale; changing λ changes the chart's statistical sensitivity, not just its display.

Every chart flags values strictly outside its limits. Optional mean-chart rules for Individuals, X-bar/R and X-bar/S flag the endpoint of nine successive points strictly on one side of the center, or six strictly increasing/decreasing points. Ties break the relevant pattern; rule windows do not cross phases. These are selected rules, not a complete Nelson or Western Electric rule suite. More rules increase false alarms. Attribute and sequential charts do not use these extra mean-chart rules.

Stability is different from specification compliance

A process can be stable and still produce unacceptable material; it can also meet today's specifications while shifting unpredictably. Control limits describe behavior. Specification limits define the engineering/customer requirement. This workspace does not overlay individual specifications on subgroup-mean or EWMA statistics, where the comparison would be misleading.

Optional LSL/USL apply to individual baseline and monitoring measurements. Observed below/above counts are reported separately; values exactly on a specification boundary count as within specification. These descriptive counts remain available without a normality assumption. No normal-model future defect forecast is made.

Capability indices require an explicit acknowledgement of a representative, stable, approximately normal process and an adequate measurement system. The tool withholds indices and intervals if the selected baseline charts/rules detect signals. Absence of a signal is not proof that the model is valid; investigate autocorrelation, mixtures and non-normality independently.

The capability within estimate is deliberately identified separately from the chart's R-bar/d₂ or S-bar/c₄ estimate. Pooling assumes a common stable mean and variance across subgroups. For individual data, the tool does not invent effective degrees of freedom for overlapping moving ranges; use Pp/Ppk intervals or collect rational subgroups.

Select 90%, 95% or 99% confidence. Spread-only bounds multiply the estimated index by the square root of a chi-square quantile divided by its degrees of freedom. For centered/one-sided indices, the approximate standard error is the square root of 1/(9N) + index²/(2df). Bounds use the appropriate standard-normal critical value. Intervals are withheld below 25 baseline measurements; fewer than 100 prompts a larger-study caution. Negative lower bounds are retained. These intervals are conditional on the model and do not certify conformity or replace an agreed acceptance criterion.

Worked process-monitoring example

1. Select X-bar/R and load its example: 35 subgroups of five measurements. The first 25 form the baseline; the final 10 contain a simulated upward shift.
2. Analyze and inspect the baseline/monitoring boundary. The shifted subgroup means produce monitoring signals while the baseline estimates stay fixed. Open the point table to see the specific reasons.
3. Review the optional specification limits of 7 and 13. Observed compliance is a different question from the detected shift. If using capability inference, read and acknowledge its assumptions; the output then includes within and overall indices and intervals.
4. Freeze the reviewed baseline, rerun the ten monitoring subgroups and verify that the limits are unchanged. Save the baseline JSON and download the report for a reproducible record.

Reports, limits and troubleshooting

Download SPC report includes settings, baseline/input data, source information, point limits and signal reasons, capability, assumptions and timestamps. Download chart CSV contains every chart point and its phase; formula-like labels are escaped. On-screen tables display the first 200 rows per chart. Charts retain every point and signal; ordinary markers are thinned on long series. Editing controls clears old results and disables result downloads until rerun.

All-zero/all-nonconforming attribute baselines and zero-variation numeric baselines are rejected because empirical limits cannot estimate the intended variation. Very large/small magnitudes may need rescaling. Mixed products, autocorrelated time series, unequal-size rational subgroups, rare-event exact charts, Laney p′/u′, multivariate charts, non-normal capability and MSA need additional methods beyond this release.

Primary method references: NIST — X-bar/R and X-bar/S; Individuals; count charts; proportion charts; EWMA; CUSUM; capability and interval limitations. Minitab — pooled degrees of freedom and capability intervals. Independent NumPy/SciPy fixtures validate the implemented formulas; they do not establish universal statistical coverage for every process.

24 · Cost of Quality & Verified Savings

Cost & Savings links operational quality losses to an improvement business case and a user-maintained benefit register. The three views remain separate: a cost reduction is not automatically a cash benefit, a forecast is not an actual result, and a “Verified” register entry records a user's review rather than this tool's certification.

Prepare the cost and production evidence

Paste CSV or a tab-separated Excel table, including the exact headers in the stated order. Use one currency across all inputs and a three-letter uppercase currency code (for example CAD). No currency conversion takes place. Use periods for decimals and omit thousands separators from numeric cells.

Periods are labels and appear in production-table order; chronological order is not inferred. Every cost line must match a production period/product. Select Read tables & update period choices after changing the tables, then choose distinct baseline and comparison periods. If an earlier selection disappears, it stays marked “not present” until corrected. Calculating also validates the tables.

Use the four categories consistently:

• Prevention: work intended to avoid quality problems, such as training and process planning.
• Appraisal: inspection, testing and other quality evaluation.
• Internal failure: losses found before delivery, such as scrap and rework.
• External failure: losses discovered after delivery, such as warranty and returns.

Total COQ adds all four categories. COPQ adds internal and external failure only. Do not include general production costs merely because they are measurable; document the quality-related scope. Avoid counting the same loss through scrap, downtime and lost contribution more than once. Released hours valued at a wage rate are not evidence that payroll fell.

Events, Pareto and comparability

Events is a nonnegative whole-number count allocated to that cost line. It is independent of Quantity. When a single incident has several cost lines, allocate its count once or use mutually exclusive event definitions; otherwise the frequency Pareto will double-count it. Training and inspection lines may have zero events. Cost and frequency rankings group by category plus cause and are sorted separately, with cumulative shares. The screen shows the top twelve causes; report/CSV include the complete ranking. Zero totals have no percentage share.

Enter the baseline rationale, valuation basis, volume unit, product-mix treatment and potential confounders in the assumptions field. Confirm complete comparable records before analysis. Missing cost rows are treated as zero only under this explicit acknowledgement. Identical cost rows are retained with a warning; legitimate repeated transactions are not silently removed.

Volume and product-mix adjustment

For each product/category, the baseline cost rate is baseline cost divided by baseline volume. Multiply that rate by the comparison-period volume for the same product, then sum across products. The result is the expected cost at baseline rates for the current volume and mix. Adjusted reduction = expected cost − actual comparison cost. Negative results indicate higher cost than that benchmark.

The period table reports COQ, COPQ, COQ per production unit, COQ as a share of revenue, and adjusted COQ reduction. Revenue share is unavailable if any product revenue is unknown or total revenue is zero. A product absent from the baseline makes the adjusted comparison unavailable rather than assigning it a zero baseline rate. Product-mix details show volume, share and baseline COPQ per unit.

This is a rate-based benchmark, not a causal model. Fixed prevention/appraisal costs may not scale with volume; review that assumption. Changes in prices, product specifications, accounting allocation, utilization, reporting completeness or measurement practice can explain an apparent reduction. Normalize those effects separately or document their treatment before attributing benefits to the project.

Projected improvement business case

The forecast form is independent of the observed cost comparison and the benefit register. Enter an annual gross cash-savings forecast, separate annual cost avoidance and released-capacity valuations, initial implementation cash cost, annual incremental cash operating cost, a 1–30-year horizon and a 0–100% annual discount rate. Do not re-enter an operating cost already deducted from a net savings forecast.

Cash net flow each year = gross cash savings − incremental operating cash cost. Year 0 contains the implementation outlay. NPV discounts each subsequent net flow at year-end and subtracts the initial cost. Cash ROI = total undiscounted net benefit divided by total implementation plus operating cash cost; it is unavailable when total cost is zero. Simple payback assumes uniform within-year cash accrual and is shown only if the initial outlay is recovered within the selected horizon. This payback convention differs from the year-end timing used for NPV.

The low/base/high scenarios vary gross annual cash savings by the chosen ±0–100% sensitivity while keeping costs fixed. They are scenarios, not probabilities or confidence intervals. Cost avoidance and capacity valuations are displayed separately and do not enter cash NPV, ROI or payback. This model has no automatic inflation, tax, financing, ramp-up, terminal-value or discount-rate recommendation. Use a discount rate consistent with the cash-flow basis. For custom cash-flow timing, use the existing Finance or Lists & Matrices workspace.

Cost analysis rejects non-finite derived results and positive cost lines that underflow to zero. Check tiny production volumes, revenues or project costs, and use consistent, practical currency and production units. A failed calculation clears prior results and disables result downloads; it does not export infinity as a valid metric or silently replace it with a missing JSON value. Legitimately unavailable metrics, such as ROI with zero total cost, remain explicitly unavailable.

Track actuals and record verification

Register types are Cash savings, Cost avoidance, Released capacity, Implementation cost and Recurring cost. Capacity Amount is a monetary valuation; record the released quantity, unit and valuation basis in Notes. Negative benefit amounts can record reversals or adverse effects. Implementation/recurring costs must be nonnegative. Cost-ledger quantity/rates are also nonnegative; reconcile credits externally into an appropriate nonnegative net line and explain the basis.

Statuses are Projected, Realized and Verified. Update the same ID when it advances; do not create one row for each stage. Projected rows may use a future period label; actual rows must reference a period in the production table. Verified records require Owner, Evidence, a valid nonfuture YYYY-MM-DD Date and Notes explaining the verification/basis. Evidence references are retained as text; the tool does not open documents, authenticate owners or approve claims.

Actual recorded totals include Realized plus Verified once. Verified is a subset, not an additional benefit. Project-wide net cash = cash savings − implementation cost − recurring cost within the relevant status scope. Totals include baseline-period initial project costs. Missing or unverified costs can overstate a verified subtotal, so pending costs are highlighted. Period tracking separates actual cash benefits, verified cash benefits, verified project costs and pending entries.

The forecast form and projected register are separate planning views and are not combined. Likewise, do not add normalized cost reductions to registered cash benefits: they may describe the same underlying improvement. Reconcile benefit ownership, scope, accounting evidence, attribution and costs with the responsible finance reviewer.

Worked manufacturing example

1. Load the fictitious manufacturing example. It has two products and Baseline, January and February periods. The production mix changes while total tonnes remain 1,500.
2. Analyze February against Baseline. Actual February COQ = CAD 16,600 and COPQ = CAD 14,700. Baseline product rates applied to February mix give expected COQ = CAD 22,500 and expected COPQ = CAD 21,000. Adjusted reductions are CAD 5,900 and CAD 6,300 respectively.
3. Review the independent forecast: CAD 50,000 annual cash benefit, CAD 8,000 annual operating cost, CAD 12,000 initial cost, three years and a 10% discount rate. Base annual net cash is CAD 42,000; NPV is approximately CAD 92,447.78; simple payback is about 0.286 years.
4. Inspect the register: CAD 3,000 cash benefits are marked Verified, CAD 4,500 are Realized but unverified, and verified implementation/operating costs total CAD 12,400. Verified net cash is therefore −CAD 9,400; actual recorded net cash is −CAD 4,900. These actual-period amounts are not the annual forecast. Released capacity is shown separately.

Reuse, save and export

For Excel/CSV reuse, import in Guided Analysis, rename/reorder its columns to match the selected target table, and copy the active dataset. Explicit exclusions and its exact category filter apply. The snapshot records source, dataset revision, source-row references and omitted-row count in the project/report. No live synchronization occurs. Editing copied text removes the old mapping for that table and records that it was edited locally.

Save the cost workspace JSON to retain all tables, settings and provenance. Restoring asks before replacing existing input, recomputes validation, clears the completeness acknowledgement and never runs automatically. Save supports valid tables with unfinished calculation settings; verified rows still require their evidence fields. The full text report retains settings, source data, all period results, cash scenarios/flows, register records and complete Pareto rankings. Results CSV provides machine-readable numeric summaries and all Pareto rows. Spreadsheet-formula-like labels are escaped. Monetary screen values round to two decimals; calculations use floating-point arithmetic.

All work is local and session-only until explicitly downloaded. Limits: 2,000 data rows and one million characters per table; 3.5 MB saved-project limit; individual numeric inputs within one trillion, with a one-quadrillion line-cost limit. Missing/invalid numbers, unknown categories, duplicate production keys and duplicate benefit IDs are rejected. Period/product output tables show up to 200 rows; full report/CSV retain the calculations. This is an analysis and recordkeeping aid, not an accounting system, approval workflow or independent assurance service.

Primary references: ASQ — cost-of-quality categories; Government Project Delivery — benefits ownership, tracking and verification. Numerical forecast fixtures use independent 40-digit Python Decimal arithmetic.

25 · Distribution Lookup Tables

This workspace reuses the simulated exam's 15 reference-data files and 18 lookup options. It provides traditional rounded tables; the separate Distributions workspace calculates probabilities and critical values for its supported distributions and parameter ranges.

Choose, find and read

1. Select Distribution Lookup Tables in the desktop tabs or mobile workspace menu.
2. Choose a table and read its explanation and probability convention. Enter the lookup keys and press Find or Enter.
3. Read the result above the grid and the highlighted row, column header and cell. Scroll inside the grid to browse other entries; keyboard users can focus the table region and use arrow keys.
4. Changing an input clears the old result and highlight. Switching workspaces preserves the current table; switching table types resets the lookup fields.

Lookup requires exact tabulated keys: it does not interpolate, round an input to a nearby grid point, truncate counts, or substitute degrees of freedom/sample size. Z inputs must be tabulated hundredths; exponential X and sigma level use tabulated tenths. Missing, invalid or non-tabulated keys show No exact match. For continuous-distribution calculations beyond this grid, select Distributions calculator. The calculators do not cover every specialized reference table.

Available tables and conventions

Scientific notation is accepted for numeric keys when it represents a valid tabulated value: for example, 1e1 means 10, not 1. Changing tables retains keyboard focus on the selected button; a late network response cannot replace a newer lookup. On narrow screens, wrapped row headings and horizontal positioning keep the selected cell visible beside the frozen row label.

Worked lookups

• Z: z = 1.96 → Φ(z) = 0.9750. Right tail = 0.0250; symmetric two-tail probability ≈ 0.0500.
• t: df = 10, α = 0.025 → t ≈ 2.228 for a two-sided 5% test.
• Chi-square: df = 10, α = 0.05 → critical value ≈ 18.307.
• F: numerator df = 4, denominator df = 20, α = 0.05 → critical value ≈ 2.866.
• Binomial PMF: n = 10, x = 2, p = 0.10 → P(X = 2) ≈ 0.19371.
• Poisson PMF: x = 3, λ = 4 → P(X = 3) ≈ 0.19537.
• Exponential: X = 2 → failure probability 0.86466 and survival probability 0.13534.

Tables load on demand from this site's shared reference files and remain cached during the page session. If loading fails, check the connection and select the table again. Table precision can round very small probabilities to zero or large probabilities to one; do not interpret these as exact impossibility or certainty. Use the Distributions calculator for more numerical precision. Source/method notes are available below the grid where recorded. No dataset upload, exam attempt or exam timer is needed.

Equations and notation

Equations are authored in LaTeX and displayed as native mathematical notation, including fractions, roots, subscripts and quantiles. The original LaTeX is retained in each equation’s MathML annotation. Programming examples elsewhere use literal input syntax so they can be copied into the calculator.