mathematics//statistics
Statistics is the machinery for reasoning from incomplete, noisy observations without pretending uncertainty disappeared, and an engineer uses it to turn logs, bench tests and alarms into numbers with an honest margin: a sensor's gain, a bearing's life, the share of alarms that are real. Probability runs from a model to the data it would produce (a sensor with gain 0.97 and 0.5 °C of noise, what will it read?); statistics makes the inverse trip, from the readings seen back to the gain and the noise that produced them.
Statistics is the machinery for reasoning from incomplete, noisy observations without pretending uncertainty disappeared, and an engineer uses it to turn logs, bench tests and alarms into numbers with an honest margin: a sensor's gain, a bearing's life, the share of alarms that are real. Probability runs from a model to the data it would produce (a sensor with gain 0.97 and 0.5 °C of noise, what will it read?); statistics makes the inverse trip, from the readings seen back to the gain and the noise that produced them.
Description summarizes the sample; inference makes claims about the process that generated it. Confusing the two is how dashboards become scientific evidence by accident.
Dispersion and joint variation are quantified by variance, covariance and the covariance matrix, and the shapes those numbers summarize are probability distributions: the normal distribution in the centre of most measurements, the exponential distribution for lives with no memory, the heavy-tailed distribution where extremes decide.
A model's assumptions are part of its output. Independence, sampling, missingness, and measurement quality often matter more than the sophistication of the estimator.
Methods such as the bootstrap quantify variability by recomputing a statistic across resampled datasets rather than demanding a convenient closed-form distribution.
The branch is organised by the question asked of the data. To get a number and its error bar, estimation (likelihood, maximum likelihood, MAP, the standard error and the confidence interval). To keep the whole spread of belief instead of one number, because a decision needs its width, Bayesian inference. To decide whether to act on a reading, hypothesis testing, which is a threshold in disguise. When a few absurd points would drag the answer, robust statistics; when the question is what causes what, causal inference, whose answer is usually to touch the system; when the data are lifetimes and many units are still alive, survival analysis. Before any of them, sample size and effective sample size say whether the data can answer at all, and selection bias asks how each row reached the table.
Four habits cover most of what an engineer needs.
Plot the data before computing anything; give intervals instead of bare numbers; work out the base rate before believing an alarm; count effective samples, never rows.
More data narrows random error. It does not automatically repair selection bias, leakage, bad labels, or a question the data never measured.