Statistics lessons for engineers

Short, practical explanations of the methods behind the app — written for people who work where the data is made.

What is Regression?

Regression finds the best formula that predicts Y from X by minimising the sum of squared errors between actual and predicted values. The result is a …

Why Transform Variables?

Real industrial data is rarely perfectly linear. LOG10, LN, inverse, or square transforms linearize curved relationships and can significantly improve…

How to Read Your Results

R² = share of variance explained (1.0 perfect, >0.85 good). Each coefficient = how much Y changes per 1 unit of that X. p-value = variable is statisti…

What are Residuals?

Residual = Actual − Predicted. Residuals near zero are ideal. Large residuals reveal measurement anomalies or unusual process conditions. Random scatt…

When to Use Hybrid Formula?

When your process behaves differently in two regimes (e.g. low vs high temperature), one formula may not fit well. Split at a threshold value → two fo…

How Reliable is My Model?

The 95% Prediction Interval (PI) shows the range where the true value will fall 95% of the time. Green PI Yellow 20–40%. Red > 40% — collect more data…

Multicollinearity & VIF

When X variables are highly correlated, the model cannot separate their individual effects — standard errors inflate and coefficients become unstable.…

What is Cook's Distance?

Not all outliers are equal. Some points have a strong influence on the regression line. Cook's Distance measures how much each row affects the model. …

What is a Q-Q Plot?

Regression assumes residuals follow a normal distribution. A Q-Q plot compares actual residuals to a perfect normal curve. Interpretation:Points near …

What is Process Drift?

Industrial processes change over time due to equipment wear, calibration drift, raw material variation, or environmental conditions. Effects:Model acc…

Prediction Interval (PI) Explained

A prediction interval shows the range where the true value is expected to fall. It reflects model uncertainty. Green PI < 20% → high confidence Yel…

Advanced Hybrid Modeling

Some processes operate in different regimes (e.g., low vs high temperature). A single formula cannot fit both regions well. Hybrid modeling:Automatica…

Model Drift Monitoring (Live Data)

When connected to live sensors, the model can monitor real-time error. Increasing MAE or residuals indicates drift. Use:Compare live MAE to training M…

Durbin-Watson: Autocorrelation Test

Autocorrelation means residuals follow a pattern in sequence — each error influences the next. This is common in time-series or ordered production dat…

Heteroskedasticity & Breusch-Pagan Test

Heteroskedasticity means residual variance is not constant — errors are small for some X values and large for others. This violates OLS assumptions an…

AIC & BIC: Model Comparison Criteria

AIC (Akaike) and BIC (Bayesian) Information Criteria measure model quality, penalizing for complexity to prevent overfitting. Lower AIC/BIC → better m…

Prediction Interval vs Confidence Interval

Both intervals measure uncertainty, but answer different questions: Confidence Interval (CI): Uncertainty about the mean prediction — how well does th…

Model Assumptions — 6-Point Checklist

Every OLS regression relies on 6 core assumptions. Violating them reduces reliability: Linearity: Relationship is linear → check Partial Regression pl…

Why Transform Variables?

Industrial data is rarely perfectly linear. Variable transforms help linearize curved relationships so the model can understand them. Common transform…

Detecting Non-Linear Relationships

Not every variable behaves linearly. Many industrial processes naturally produce curved patterns that a straight-line model cannot capture. Signs of n…

Outliers and Their Impact

Outliers are data points that behave very differently from the rest. They may come from sensor noise, operator error, or real process anomalies. Why o…

When to Use Hybrid Formula

Hybrid Formula combines a regression model with a rule-based threshold adjustment, blending statistical math with process engineering knowledge. Use H…

How to Read Your Results

After running regression, the Results tab shows several key metrics: R² — how well the model fits (0 to 1; higher is better)Coefficients — how much ea…

Why More Samples Improve Accuracy

Industrial processes have natural variability. More samples help the model learn the full range of behavior: Normal operating rangeRare events and pro…

SPC — Control Charts (Is the Process Stable?)

A control chart watches a measurement over time and asks one question: is the process behaving normally, or did something change? The I-chart (Individ…

Process Capability (Cp / Cpk)

Control limits tell you if a process is stable. Capability tells you if it's good enough for your specification. You enter your spec limits (LSL = low…

t-Test (Are Two Groups Really Different?)

When you have two groups (two machines, two shifts, before vs after a change) and want to know if their averages truly differ — not just by luck — you…

Pareto Analysis (The 80/20 Rule)

Most problems come from just a few causes. A Pareto chart ranks causes from most to least frequent (bars) and draws a cumulative line (red). Where the…

Data Point Evidence (Trustworthy Data)

A model is only as good as its data. Real measurements sometimes go wrong: a defective sample, an interrupted reading, a miscalibrated instrument. Sen…

Gage R&R & MSA (Can You Trust Your Measurements?)

Before trusting any data, ask: is my measuring reliable? Gage R&R (a Measurement System Analysis) splits your total variation into two parts:Measureme…

DOE — Design of Experiments (What Really Matters?)

Instead of changing one factor at a time, DOE varies several factors together (each at a low and a high level) so you learn their effects efficiently …

Cross-Tab — Reading Significance Letters

A cross-tab counts how two label columns intersect: defect type by shift, reject reason by supplier, answer by department. The percentages alone are n…

Scales — Top-2-Box, Reverse Coding and NPS

A 1–5 rating question is usually reported as top-2-box: the share of people who picked one of the two best answers. It is blunt on purpose — an averag…

Coding Open-Ended Answers

Free-text answers and complaint records cannot be counted until they are grouped. Traditionally somebody reads a few hundred of them and sorts them in…

Multi-Response Questions

"Which of these did you experience? Tick all that apply" produces a cell holding several answers at once — Noise;Dust;Heat. The row is still one perso…

Which control chart should I use?

SenSight offers five chart types and the choice is not cosmetic — the wrong chart either misses a real problem or cries wolf. I-MR — you measure one p…

Cp, Cpk, Pp, Ppk — which one do I report?

Four indices, two questions. Cp vs Cpk — Cp asks "is the spread narrow enough to fit the spec?", Cpk asks "and is it also in the right place?". A perf…

Which predictors should I keep?

You have eight columns and no idea which belong in the model. Two classical tools answer that — and both come with a warning. Best subsets fits every …

When a straight line is the wrong model

Ordinary regression assumes two things people rarely check: the response can take any value, and its spread is the same everywhere. Two very common ki…

I know which way to move — but where do I stop?

A two-level factorial answers the first question well: raise the temperature, yield goes up. It cannot answer the second, and the reason is geometric …

Two goals that pull in opposite directions

A response surface finds the peak of one response. No real process has one response. You want the yield up and the scrap down, the hardness on target …

The setting with the best average is not always the best setting

Two machine settings produce the same average tensile strength. One of them gives you the same number every day; the other swings with the humidity, t…

When the ingredients must add up to 100%

Concrete, an alloy, a paint, a feed mix: you are choosing proportions, and proportions sum to 1. This one constraint breaks the ordinary machinery of …

Thirty columns that are really about four things

Open a process export and you will find thirty measurements side by side. Look closer and most of them move together: three thermocouples on the same …

A forecast without a range is not a forecast

Someone asks what next month looks like and you give them a number. They write it down, plan around it, and it is wrong — as every forecast is. The nu…

When measuring destroys the part

A gauge study asks a simple question: when the number changes, is that the part changing or the measurement changing? The standard answer is to have s…

Three decisions people make by feel

Most quality decisions come with arithmetic attached, and three of them are almost always made without it. Not because people are careless — because t…

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