Engineering & Statistics Calculator

Distribution Lookup Tables

Browse the reference tables used in the simulated exams. Choose a table, enter your lookup keys, and select Find to highlight the matching row, column and cell.

Lookup tables manual →

Table values are rounded and cover a fixed grid. This workspace requires exact tabulated keys; it does not interpolate or substitute degrees of freedom. For values outside the tables, use the .

Choose a reference table

1. Load your data

Paste a table from Excel or CSV (first row as headers)

Use a period for decimals.

Import Excel (.xlsx), CSV or TSV

Files stay in this browser. Nothing is uploaded to a server. Up to 10 MB, 10,000 rows and 40 columns.

Or try a fictitious sample

Best linear model

Every model that becomes a straight line after transforming x and/or y was checked: the 64 ladder-of-powers combinations, logit, Arrhenius, Box–Cox and Box–Tidwell. The pick passes both residual checks where possible and, among near-ties, has the fewest parameters.

Recommended

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Runners-up
Model	ΔAICc	R² (y)	Checks

Transformed scale

2. Ranked models

Rank by Show

#	Model	Family	k	AICc	ΔAICc	Weight	R² (y)	CV RMSE	Checks	View

Models that could not be fitted

3. Selected model

Data on the original scale

Selected model
Next two by rank

Residuals vs fitted (fitting scale)

Fitting scale

Parameters
Name	Estimate	Std. error

Predict

x value

For models fitted on a transformed scale, the plain back-transform is the median; the smearing estimate is the mean. Intervals are 95% prediction intervals for a single new observation. GLM rows give the mean only.

How the ranking works

Every model is scored on y itself. A model fitted to \(g(y)\) assumes normal errors on that scale. Its log-likelihood for the original y adds the Jacobian of the transformation, which is what makes AICc comparable across \(y\), \(\ln y\), \(\sqrt{y}\) and the rest:

\[ \ell_{y} = -\frac{n}{2}\left[\ln\!\left(\frac{2\pi\,\text{SSE}_{g}}{n}\right) + 1\right] + \sum_{i=1}^{n} \ln\left|g'(y_i)\right| \]

where \(\text{SSE}_{g}\) is the error sum of squares on the fitting scale and \(n\) is the number of points.

\[ \text{AICc} = -2\ell_{y} + 2k + \frac{2k(k+1)}{n-k-1}, \qquad w_m = \frac{e^{-\Delta_m/2}}{\sum_j e^{-\Delta_j/2}} \]

where \(k\) counts every estimated parameter including the error variance, \(\Delta_m\) is a model's AICc minus the lowest AICc, and \(w_m\) is its Akaike weight. Models within \(\Delta < 2\) are effectively tied: prefer the simplest one, or the one with a physical meaning. The Poisson GLM has a discrete likelihood, so its AICc is shown but not ranked against the continuous models; compare it by cross-validated RMSE.

Cross-validated RMSE refits each model on 9 of 10 folds and predicts the held-out tenth on the original y scale, so overfitting costs it directly. R² (y) scores back-transformed median predictions against the raw y, exactly as in the lesson.

Checks apply the lesson's two screens to each model's residuals on its fitting scale: a partial F-test for leftover curvature (squared and cubed fitted terms) and a test of whether the absolute residuals trend with the fitted value. A model that wins on AICc but fails a check is fitting the wrong shape or the wrong error structure.

Model families tried

Ladder of powers: all 64 combinations of \(x^{p}\) and \(y^{q}\) for \(p, q \in \{-2, -1, -\tfrac12, 0, \tfrac12, 1, 2, 3\}\), which include linear, exponential, power law, logarithmic, reciprocal and Hall–Petch forms.
Logit of proportions or percentages against \(x\) and \(\ln x\).
Arrhenius \(\ln y\) and \(y\) against \(1/T\) in kelvin, when x is a temperature.
Box–Cox power on y estimated by maximum likelihood, against five x rungs.
Box–Tidwell \(y = a + b\,x^{\alpha}\) with \(\alpha\) estimated.
Polynomials (quadratic and cubic).
Nonlinear least squares: exponential and power law with additive error, exponential approach to a floor or ceiling, and the hyperbolic-tangent transition curve.
GLMs with a log link: Gamma for positive skewed y and Poisson for counts, against \(x\) and \(\ln x\).

Use judgement. The best statistical fit is not automatically the right model. Prefer a model whose form matches the physics, check it on new data, and do not extrapolate a cubic or a back-transformed exponential far beyond the observed range.

CONNECT QUALITY TO BUSINESS RESULTS

Cost of quality and COPQ calculator

Analyze prevention, appraisal, internal failure and external failure costs. Compare improvement scenarios and track documented benefits without treating every cost reduction as cash savings.

Learn the method in the Cost of Poor Quality lesson.

Cost & savings manual →

1 Establish cost & volume evidence
2 Compare the improvement case
3 Track actuals & verification

Project inputs

Restore cost workspace JSON

Currency code (one currency)

Production volume unit

1 · Quality cost ledger

Enter the headers below. Categories: Prevention, Appraisal, Internal failure, External failure. Line cost = Quantity × UnitCost. Use quantities such as scrap tonnes or rework/downtime hours and matching rates. Events are separately allocated counts, not quantities.

Cost ledger CSV / Excel paste

2 · Production volume & revenue

One row per period/product. Positive volume in one consistent production unit. Blank revenue means unknown; zero means zero. Periods follow table order. A baseline rate for each product adjusts comparisons to the current volume and product mix.

Production CSV / Excel paste

Import from Guided Analysis

Import Excel/CSV there first. Rename/reorder the columns to exactly match the target table headers. Explicit exclusions and the exact category filter apply. This copies a snapshot, with provenance in the report.

Target table

Directly entered tables.

Baseline period

Comparison / Pareto period

Baseline, valuation, product-mix & attribution assumptions I have reviewed complete, comparable records, consistent valuation and non-overlapping event counts. Missing cost rows represent zero within this scope.

3 · Projected improvement business case

Standalone annual forecast, not calculated or verified from the cost reduction. Year 0 implementation outlay; constant annual benefits/costs over the horizon. Only cash savings enter cash NPV, ROI and payback.

Projected annual cash savings

Annual cost avoidance (noncash here)

Annual released capacity valuation (noncash)

Initial implementation cash cost

Annual incremental cash operating cost

Horizon (whole years, 1–30)

Annual discount rate (%)

Cash benefit sensitivity (±%)

Low/base/high scenarios vary gross cash benefit only; costs stay fixed. Use a rate consistent with your nominal/real cash-flow basis. No tax, inflation, terminal value or borrowing model is applied.

4 · Benefit & cost verification register

Optional. Types: Cash savings, Cost avoidance, Released capacity, Implementation cost, Recurring cost. Status: Projected, Realized, Verified. Keep one unique ID per benefit/cost and update its status; do not duplicate it across stages. Amounts are period totals in the selected currency; capacity is a noncash valuation.

Benefit register CSV / Excel paste

Verified entries require an owner, evidence reference, valid nonfuture date (YYYY-MM-DD) and basis/verification notes. Negative benefit amounts record reversals; cost amounts must be nonnegative. Verification is user-recorded, not authenticated or approved by this tool.

Session-only, local calculations. Up to 2,000 rows / 1 MB per table. Save before closing. Do not add ledger cost reductions, forecasts and register benefits together.

Quality economics

Load an example or enter your cost and production tables.

UNDERSTAND VARIATION · MONITOR CHANGE

Control charts and process capability

Choose a chart for your sampling design, establish a baseline, monitor new observations against fixed limits, and distinguish process stability from specification compliance.

Review the control charts lesson before interpreting signals or assessing capability.

Advanced SPC manual →

1 Choose the sampling model
2 Review and freeze a baseline
3 Investigate signals and capability

Data & chart design

Control chart

Reuse data from Guided Analysis

Import Excel/CSV and edit it in Guided Analysis first. Choose columns in the order required above. Current explicit exclusions and the exact category filter are applied; blanks are rejected. The copy is a snapshot.

Select “Show active dataset” to see its columns.

Column numbers, in chart order (for example 2,3)

Time-ordered numeric rows (no headers)

Separate columns with commas, tabs or spaces. Up to 10,000 values and 250,000 characters. Missing cells are rejected; blank lines are ignored. Preserve chronological order.

Pasted measurements

Initial rows used to estimate baseline

Only these first rows estimate limits. Later rows are monitoring observations. Include at least 20 representative baseline rows when possible.

No frozen baseline. Estimate from the first rows.

Restore baseline JSON

Freezing keeps only the later monitoring rows in the editor; baseline data remain in the snapshot. With a frozen baseline, paste the complete monitoring sequence each time. Save before closing; there is no automatic storage.

Also flag 9 points on one side and 6-point trends on the mean chart. I confirm chronological order, one process, independent sampling (or rational subgroups), and a suitable measurement system.

Specification compliance & capability

Optional engineering limits apply to individual measurements, not subgroup means. Capability estimates use the baseline only.

Lower specification (optional)

Upper specification (optional)

Capability confidence levelFor capability inference, I have reviewed a representative, stable, approximately normal process and the measurement system.

Process evidence

Choose a chart and load its example to get started.

FROM DATA TO A DECISION

Start with your question.

Bring your data, choose what you need to learn, and work through the evidence with clear assumptions and next steps.

Guided workspace manual →

1 Bring & review data
2 Define your question
3 Interpret & take action

1 · Import or restore data

Paste a table from Excel or CSV

Text separator

First row contains column names

Import Excel (.xlsx), CSV or TSV

Files stay in this browser. Up to 10 MB per file; 10,000 rows, 40 columns and 100,000 cells per dataset. Use a period for decimals. Older .xls files must be saved as .xlsx or CSV.

Restore a saved workspace (.json)Try a worked dataset

Add a table or try a worked example. Up to five datasets can share this session.

2 · Define your question

What are you trying to learn? Response / measurement column

Confidence level

Response unit (optional)

Filter rows and handle missing values

Include only one category Use complete rows only and list every missing-value exclusion.

Only blanks in the selected analysis columns are treated as missing. Invalid numbers block analysis. Paired rows are excluded together.

Guidance follows your question and study design. It does not automatically certify assumptions or choose a method using a normality-test p-value.

3 · Understand the evidence

Import data, select your columns and review the suggested method.

The report includes inputs, settings, exclusions, interpretation and next steps. Full results become unavailable when inputs change.

Distribution

Question

Region

Test

Alternative hypothesis

Significance level α

Analyze your data

Analysis

Paste numeric data

Decimal point: period. Separate cells with tabs, commas, semicolons, or spaces. No headers or thousands separators. Blank lines are ignored; invalid or empty cells are reported.

Confidence level

Significance α

Calculations run in your browser. This workspace does not upload or save your dataset.

Results

Choose an analysis, paste data, or load an example.

Methods, assumptions & references

Intervals for a mean use Student’s t. Regression fits ordinary least squares with an intercept; residual standard error uses n − 2 degrees of freedom. ANOVA is the classical equal-variance F test. Chi-square uses Pearson’s statistic without continuity correction and reports every expected cell count.

Capability uses s for Pp/Ppk and MR̄ ÷ 1.128 for Cp/Cpk. I-MR limits are estimated from the entered sequence; only points beyond three-sigma limits are flagged. Review stability, independence and distribution shape before interpreting capability. Specification limits are engineering requirements, not control limits.

Graph & explore

Graph mode Angle unit Expressions (one per line, up to 10)

Use x, pi, sin(x), abs(x), sqrt(x), log(x), or a piecewise expression such as x < 0 ? -x : x.

X min

X max

Y min

Y max

Table settings (function mode)

Table startTable stepRows (1–200)

Graph window

Draw the example or enter your own expressions.

Trace curveTrace sampled points (arrow keys supported)

Numerical analysis (Cartesian functions)

Operation on first expression

a / lower bound

b / upper bound

Zeros need a sign-changing bracket. A minimum/maximum search assumes one extremum within the interval. Plot sampling can miss narrow features; check the window and numerical result.

Math workspace

Angle unitExpression, list, matrix, or assignment

Lists: L1 = [1,2,3]. Matrices: A = [1,2;3,4]. Complex: sqrt(-1), (2+3i)*(1-i). Units: 25.4 mm to inch. Last result: Ans. Variables are shared with graphing and programs for this session.

Worked exampleSearch supported functions

Calculation tape

Evaluate expressions using the examples or function catalog.

Syntax & numerical tools

Use * for multiplication and ^ for powers. ln(x) and log(x) are natural logarithms; log10(x) is base 10. Complex results use a + bi. arg(z) returns radians. RAD/DEG applies to trigonometric functions, their inverses, atan2, and the nDeriv/fnInt/solve/seq helpers. Complex arg/angle remains in radians.

nDeriv("x^3",2), fnInt("sin(x)",0,pi), solve("x^2-2",0,2), seq("n^2",1,10), rref([1,2,3;2,1,4]).

Matrix/list indexing and user-defined functions are not enabled. Use whole-array operations. Large operations are bounded; a timeout resets workspace variables to keep the interface responsive.

Time value of money

Solve for

N (payment periods)

I% (annual nominal)

PV

PMT

FV

Payments per year

Compounds per year

Payment timing

Money received is positive; money paid is negative. N counts payment periods, not years. The selected unknown is ignored until solved.

Financial results

Cash flows & rate conversion

In Lists & Matrices, use npv(0.10,[-1000,400,400,400]) for a 10% per-period discount rate. The first cash flow occurs at time zero. Use irr([-1000,400,400,400],0,1) for the decimal periodic IRR within a bracket. IRR can have multiple roots or no root.

eff(6,12) converts 6% annual nominal interest, compounded monthly, into an annual effective percentage. nom(6.1678,12) performs the reverse conversion. TVM uses an annual nominal percentage; NPV/IRR use decimal rates per cash-flow period.

Calculation programs

Program sourceAngle unit

Programs run locally in an isolated worker. Maximum 500 lines and 10,000 execution steps. A timeout resets shared variables.

Program output

Supported programming subset

Assignments (value→A, value->A, or A=value), Disp expression, ClrHome, For(I,start,end,step), While condition, If condition / Then / Else / End, and Stop. Every control block requires End, including If.

Use the math-workspace function catalog. This is a documented calculation-language subset, not a TI-BASIC emulator: device I/O, key events, labels/Goto, graphical commands, tokenized .8xp files and installed TI apps are not supported.

SHIFT ALPHA hyp STO DEG ENG S⇆D M: 0

0

History tape

Click any entry to reuse it. Ans holds the last result.

Data / STAT mode

Data format Numeric data

Engineering & Statistics Calculator

Distribution Lookup Tables

1. Load your data

Explore transformations

Best linear model

–

2. Ranked models

3. Selected model

Predict

How the ranking works

Cost of quality and COPQ calculator

Project inputs

Quality economics

Control charts and process capability

Data & chart design

Process evidence

Start with your question.

2 · Define your question

3 · Understand the evidence

What to do next

Analyze your data

Results

Graph & explore

Graph window

Math workspace

Calculation tape

Time value of money

Financial results

Calculation programs

Program output

History tape

Data / STAT mode