statistics basics

Statistics is the discipline of turning raw data into trustworthy conclusions, and it is the daily language of a semiconductor fab and design lab where measurements are taken by the millions and decisions hinge on whether a difference is real. Where probability reasons forward from a known model to the likelihood of observations, statistics reasons backward from observed data to the unknown process that produced it, so a process engineer who sees a threshold-voltage spread must infer the underlying mean and variance, a yield engineer who sees a defect count must estimate the true defect density, and a designer who compares two SRAM cell variants must decide whether one is genuinely faster. This document develops statistics basics from the ground up through the lens of semiconductor engineering, beginning with how to describe data, moving to how to draw samples that represent a population, then to estimation and hypothesis testing, and finally to the regression, experimental-design, and resampling methods that modern engineers actually run. An engineer fluent in these foundations can read a fitted model, challenge a control limit, and ask the right questions of a data scientist, which is the difference between a tool user and a decision maker. **The population is the complete set of objects of interest, while a sample is the subset actually measured.** A population parameter is a fixed but usually unknown number that describes the entire population, such as the true mean threshold voltage of every transistor on a production wafer, while a sample statistic is a number computed from the measured subset that serves as an estimate of the parameter. The central goal of inferential statistics is to move from the statistic, which is known, to the parameter, which is not, and to state how much uncertainty remains after that move. In a fab, the population is effectively all dies of a lot, all wafers of a run, or all devices that will ever be manufactured under a recipe, which is why a sample drawn today must stand in for millions of future parts. The distinction between parameter and statistic is the first and most important labeling act in any analysis, because every subsequent formula and interpretation depends on knowing which quantity is fixed and unknown and which is random and observed. A statistic is a random variable, since it changes from sample to sample, while a parameter is a fixed constant of the population, and confusing the two leads to misstated uncertainties and misplaced confidence. When a yield engineer reports that a wafer's defect density is a particular number, she is reporting a statistic that estimates the true process defect density, and the gap between them is exactly the sampling uncertainty that inferential tools are built to quantify. **Descriptive statistics summarize a dataset with a few numbers that capture its center and its spread.** The three measures of central tendency are the mean, the median, and the mode, with the arithmetic mean $\bar{x} = \frac{1}{n}\sum x_i$ being the balance point of the data, the median being the middle value that splits the sorted data in half, and the mode being the most frequent value. The mean is sensitive to outliers while the median is robust to them, so a skewed distribution such as a particle count or a time-to-failure is often better summarized by the median than by the mean. The spread is measured by the range, the interquartile range, and the variance $s^2 = \frac{1}{n-1}\sum (x_i - \bar{x})^2$, whose square root is the sample standard deviation $s$. Reporting the mean and standard deviation together is the default summary for a roughly symmetric, normal-like process such as a film thickness, while a median and interquartile range better describe a skewed one. It is a common error to report only the mean, because the same mean can arise from a tight, well-centered process or from a wide, sloppy one, and the spread is what controls whether a process fits its specification. The sample variance uses $n-1$ in the denominator rather than $n$ because the deviations are measured from the sample mean, which itself must be estimated, and this small correction, called Bessel's correction, makes the sample variance an unbiased estimate of the population variance. An engineer who quotes only the average thickness without its standard deviation is hiding exactly the information that determines yield and capability. **The histogram and the box plot are the two graphics that reveal the shape of a distribution at a glance.** A histogram partitions the range of the data into bins and draws a bar for each bin whose height is the count, exposing the shape, the center, the spread, and any gaps or outliers, and the choice of bin width strongly affects the impression the histogram gives, with too-few bins hiding detail and too-many bins showing only noise. A box plot displays the median as a line, the interquartile range as a box, and the tails as whiskers, with individual points flagged beyond the whiskers as potential outliers, making it the ideal tool for comparing several groups side by side. The quantile-quantile plot, which compares the sorted data to the quantiles of a reference distribution such as the normal, is the standard diagnostic for whether data follow a normal distribution, with points hugging the straight line when the fit is good. Together these graphics let an engineer see a dataset before trusting any single-number summary. **Data come in types that determine which statistical methods are legitimate to apply.** Categorical data consist of labels or categories, such as the defect class or the bin of a failed die, while numerical data consist of quantities, such as a critical dimension or a measured resistance, and numerical data are further split into discrete counts and continuous measurements. The four levels of measurement run from nominal categories with no order, through ordinal categories with an order but uneven spacing, to interval scales with equal spacing but no true zero, and finally to ratio scales with a meaningful zero, and the level of measurement restricts which statistics are sensible, since computing a mean of nominal labels is meaningless. In a semiconductor context, the distinction between a measured thickness and a categorical pass-fail outcome dictates whether a t-test or a proportions test is appropriate. Choosing the right tool for the data type is the practical gateway to valid statistics. **Random sampling is what makes a small sample representative of a large population.** In a simple random sample, every member of the population has an equal chance of being selected and selections are independent, so the sample statistics estimate the population parameters without systematic bias. Stratified sampling divides the population into groups and samples from each, guaranteeing representation of every stratum, which matters when a wafer has known within-wafer zones or a lot has known within-lot wafers that should each appear. Cluster sampling and systematic sampling reduce cost when the population is physically spread out, but they introduce extra dependence that must be handled in the analysis. The principle that underlies all of sampling is that randomization removes the bias of human selection, so an engineer who wants to know the mean thickness of a lot must measure dies selected at random, not the convenient ones at the wafer edge. **The sampling distribution describes how a statistic would vary across many repeated samples, and it is the engine of inference.** Because a sample statistic is itself a random variable that changes from sample to sample, it has a distribution called the sampling distribution, whose standard deviation is the standard error, and the standard error of the mean is $\sigma/\sqrt{n}$, so the uncertainty of a sample mean shrinks as the square root of the sample size. The central limit theorem states that the sampling distribution of the mean is approximately normal regardless of the shape of the population, provided the sample size is large enough, and this is what licenses normal-based inference for means even when the underlying data are skewed. The distinction between the standard deviation of the data and the standard error of the mean is one of the most common confusions in practice, and getting it right is essential to interpreting any confidence interval. When an engineer quotes a critical-dimension measurement as a value plus or minus an uncertainty, the uncertainty is almost always a standard error. **Point estimation selects a single best guess for an unknown parameter, judged by bias and variance.** An estimator is unbiased when its expected value equals the true parameter, so that on average it neither overestimates nor underestimates, and it is consistent when it converges to the parameter as the sample size grows, while among unbiased estimators the efficient one has the smallest variance. The method of maximum likelihood chooses the parameter that makes the observed data most probable, and under regularity conditions maximum likelihood estimators are consistent and asymptotically normal, with the Cramér-Rao lower bound setting the smallest possible variance of any unbiased estimator. The trade-off between bias and variance is fundamental: a deliberately biased but low-variance estimator can have smaller mean squared error than an unbiased but high-variance one, a lesson that drives modern shrinkage and regularization methods. In practice, the mean of a normal sample and the sample proportion of a binomial sample are the two most common point estimators. The sample mean $\bar{x}$ is an unbiased and consistent estimator of the population mean, and it is the maximum likelihood estimator for the normal model, which is why it appears everywhere in process measurement. The sample proportion $\hat{p} = x/n$ estimates a population proportion such as a pass rate or a yield, and it too is unbiased and consistent, with its variance $p(1-p)/n$ largest near the middle of the range. Because a point estimate alone carries no information about precision, every responsible analysis pairs the estimate with a standard error or a confidence interval, so that the reader can see not just the best guess but how much the guess could be off. **A confidence interval converts a point estimate into a range that plausibly contains the true parameter.** A 95 percent confidence interval for a normal mean is $\bar{x} \pm t \cdot s/\sqrt{n}$, where the multiplier comes from the t distribution with $n-1$ degrees of freedom for small samples and approaches the normal multiplier for large ones, and it is constructed so that 95 percent of intervals built this way across repeated sampling contain the true mean. The width of the interval is determined by the sample size, the variability, and the confidence level, and it communicates the precision of the estimate far more honestly than a bare point value. The confidence level is a property of the method, not of a single interval, so a particular interval either contains the parameter or it does not, and the correct statement is about the repeated-sampling procedure. Reporting a yield or a capability index with its confidence interval tells a reviewer how much the number could move with more data. For a proportion, the 95 percent confidence interval is approximately $\hat{p} \pm 1.96\sqrt{\hat{p}(1-\hat{p})/n}$, so a yield measured on a small sample carries a wide interval and must not be over-interpreted as the true yield. The trade-off is direct and worth internalizing: to halve the width of a confidence interval one must quadruple the sample size, because the standard error falls only as the square root of $n$. A capability index quoted without its interval, or a yield quoted from a handful of wafers, invites a decision on evidence that is far weaker than it appears. **Hypothesis testing frames a research question as a choice between a null and an alternative hypothesis.** The null hypothesis $H_0$ is a claim of no effect or no difference, such as that two processes produce the same mean thickness, while the alternative hypothesis $H_1$ is the claim the study is designed to detect, such as that one process is thicker. A test statistic computed from the data measures how far the evidence departs from what the null predicts, and the decision to reject the null is made when the statistic falls in a region that would be unlikely if the null were true. A type I error rejects a true null with probability $\alpha$, and a type II error fails to reject a false null with probability $\beta$, with the power to detect a real effect equal to $1-\beta$. The significance level $\alpha$, conventionally 0.05, is the acceptable false-alarm rate, and the design must ensure enough samples to make the power adequate. **The p-value is the probability of observing data at least as extreme as the actual data if the null hypothesis were true.** A small p-value means the observed result would be surprising under the null and therefore argues against it, and the test rejects the null when the p-value falls below the significance level. The p-value is not the probability that the null is true, nor the probability that the result is a false positive, and it is itself random, fluctuating across replications, so single p-values near the threshold should be treated cautiously. Multiple testing inflates the chance of a false positive, which is why methods such as the Bonferroni correction and the Benjamini-Hochberg false-discovery-rate procedure adjust p-values when many tests are run at once. In a fab comparing many parameters between two recipes, an engineer must account for the fact that one significant result in twenty is expected by chance alone. The significance level $\alpha$ is a false-alarm rate per test, so if an engineer runs two hundred tests at the 0.05 level, roughly ten false positives are expected even when nothing is really different, and naive reporting of the few significant results is an error known as p-hacking. Controlling the family-wise error with a stricter threshold, or controlling the false discovery rate so that a stated fraction of reported findings are expected to be true positives, keeps the conclusions honest, and any adjustment for multiple comparisons must be decided in advance rather than after seeing the results. The larger lesson is that a single small p-value is weak evidence on its own; the strength of a finding comes from a pre-specified hypothesis, an adequate sample, a controlled design, and reproducibility on independent data. **The t-test compares the means of one or two groups and is the workhorse of mean comparison.** A one-sample t-test asks whether the mean of a sample differs from a known target, a two-sample t-test asks whether two independent groups have different means, and a paired t-test asks whether two related measurements on the same subjects differ, such as a measurement before and after a process change on the same wafers. Each t-test computes a t statistic as the observed difference divided by its standard error and refers it to the t distribution, with the number of degrees of freedom depending on the sample size and whether the group variances are pooled. The t-test assumes the data are approximately normal and the observations independent, and it is reasonably robust to modest departures from normality when the sample is large. When a design engineer compares the delay of two SRAM cell variants measured across many instances, the paired or two-sample t-test decides whether the difference is real. The t distribution, introduced by William Gosset writing under the pen name Student while at Guinness, is heavier-tailed than the normal distribution to reflect the extra uncertainty of estimating the variance from a small sample, and its shape depends on the degrees of freedom, converging to the normal as the sample grows. For a two-sample t-test the degrees of freedom account for whether the two group variances are assumed equal, with Welch's adjustment allowing unequal variances and yielding a fractional degree of freedom. The choice between pooled and Welch versions matters when the two groups have quite different spreads, as often happens when comparing a mature process with a new one, and modern software defaults to the more conservative Welch version. **Analysis of variance (ANOVA) extends mean comparison to three or more groups and is built on partitioning total variation.** Analysis of variance tests the null hypothesis that several group means are all equal by comparing the variation between group means to the variation within groups, and the F statistic is the ratio of these two mean squares, with a large F indicating the groups differ more than would be expected by chance. A one-way analysis of variance compares a single factor with several levels, a two-way analysis of variance handles two factors and their interaction, and a factorial design generalizes the idea to many factors at once. When the F test rejects the global null, follow-up comparisons locate which pairs of groups differ, while controlling the overall error rate across the comparisons. In process development, analysis of variance is how an engineer learns whether a factor such as temperature or pressure has a real effect on a response. **The chi-square test compares observed counts with expected counts and handles categorical data.** A chi-square goodness-of-fit test checks whether observed frequencies match a claimed distribution, such as whether defect counts follow the Poisson model, and a chi-square test of independence checks whether two categorical variables are associated, such as whether the defect type is independent of the production shift. The test statistic sums the squared differences between observed and expected counts divided by the expected counts, and it is referred to the chi-square distribution whose degrees of freedom depend on the number of categories. The test is valid only when expected counts are not too small, and the data must be counts rather than continuous measurements. For categorical quality data, the chi-square test is the standard way to detect whether a shift in the defect mix is real or just sampling noise. **Correlation measures the strength and direction of the linear association between two variables.** The Pearson correlation coefficient $r$ ranges from negative one to positive one, with zero indicating no linear relationship, and it is computed from the covariance of the two variables standardized by their standard deviations, while the Spearman rank correlation replaces the data by their ranks and therefore detects monotonic relationships even when they are not linear. Correlation does not imply causation, and it can be inflated or deflated by outliers, restricted ranges, and a third lurking variable, so a strong correlation between two process parameters does not by itself identify which one drives the other. The sample correlation inherits sampling variability, so its confidence interval and the hypothesis test that it is nonzero should always accompany the point value. In metrology, correlation between a fast measurement and a slow reference measurement is what validates a virtual metrology model. **Linear regression models a response as a linear function of one or more predictors by minimizing squared error.** The simple linear regression model $y = \beta_0 + \beta_1 x + \epsilon$ assumes an independent normal error with zero mean and constant variance, and the least squares estimates choose the line that minimizes the sum of squared residuals. The coefficient of determination $R^2$ is the fraction of the variance of the response explained by the model, and the standard errors of the coefficients, their t-statistics, and their p-values indicate which predictors are significant. The assumptions of linearity, independence, homoscedasticity, and normality of residuals must be checked with residual plots, because a curved pattern or a funnel shape signals a violated assumption. Regression is everywhere in the fab, from calibrating a film-thickness model to a metrology measurement to fitting a delay model to simulated results. **Multiple regression fits a response to several predictors and is the foundation of empirical models.** The multiple regression model $y = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k + \epsilon$ estimates each predictor's effect while holding the others fixed, and the coefficients and their standard errors tell an engineer which factors matter and by how much. Multicollinearity, in which predictors are strongly correlated, inflates the standard errors and makes individual coefficients unstable even when the overall fit is good, so correlated predictors must be examined and possibly combined or removed. Model selection among predictors uses criteria such as adjusted $R^2$, Akaike's information criterion, and cross-validated prediction error, balancing fit against complexity. In process modeling, multiple regression turns a set of recipe and measurement variables into a predictive equation that is cheap to evaluate and easy to interpret, provided the model is not extrapolated outside the region of the data. **Nonparametric methods relax the normality assumption and use ranks instead of raw values.** When data are strongly skewed, ordinal, or few in number, the assumptions of the t-test and analysis of variance may fail, and rank-based tests offer a valid alternative. The Wilcoxon signed-rank test replaces the paired t-test, the Mann-Whitney U test replaces the two-sample t-test, and the Kruskal-Wallis test replaces one-way analysis of variance, each using ranks so that only the ordering of the data matters rather than their numerical spacing. These tests lose some power when the normality assumption truly holds, but they protect against incorrect conclusions when it does not, making them a robust default for skewed data such as particle counts. A practitioner should decide on the test based on the data's distribution and the design, not on convenience. **Statistical process control monitors a process over time to separate routine variation from assignable causes.** A control chart plots a statistic such as the sample mean or a count over time with a center line and upper and lower control limits usually set at three standard deviations, and points within the limits in a random pattern indicate a process in control that should be left alone. A point beyond the limits, or a systematic run such as several points on one side of the center line, signals a special cause that warrants investigation and correction. The process capability indices $C_p$ and $C_{pk}$ then compare the process spread to the specification width, quantifying how well the process fits its tolerances. Walter Shewhart's insight at Bell Labs remains the foundation of modern fab monitoring, where thousands of metrology parameters are charted continuously and an excursion trips a hold. **Design of experiments plans data collection to estimate factor effects with maximum efficiency and minimum bias.** A designed experiment varies multiple factors in a structured way instead of changing one factor at a time, so that main effects and interactions can be estimated from a modest number of runs. A full two-level factorial in $k$ factors needs $2^k$ runs, a fractional factorial trades some resolution for far fewer runs, and response surface methodology fits a quadratic model near an optimum to locate the best setting. Randomization of run order and blocking of nuisance factors ensure that the estimated effects are not confounded with drift or background variation. When a fab wants to find the etch recipe window, a designed experiment on gas flow, power, and pressure reveals which factors matter and how they interact. **Multivariate statistics analyzes many correlated variables together to expose hidden structure.** Principal component analysis diagonalizes the covariance matrix of many measured variables and projects the data onto the directions of greatest variance, reducing a high-dimensional metrology dataset to a few interpretable components. Factor analysis and clustering methods group similar observations or similar variables, revealing wafer zones, recipe families, or defect patterns that a univariate look at each variable would miss. Hotelling's T-squared statistic generalizes the t-test to the multivariate mean, and multivariate control charts monitor many parameters simultaneously so that a subtle joint shift that no single chart would catch is detected. In advanced process control and metrology, multivariate methods turn a wall of correlated sensors into a small set of meaningful signals. **Resampling methods such as the bootstrap estimate uncertainty by resampling the data itself.** The bootstrap, due to Bradley Efron, draws many samples with replacement from the observed data, recomputes the statistic of interest each time, and uses the distribution of these recomputed values to estimate the standard error and confidence interval without strong distributional assumptions. The jackknife, which leaves out one observation at a time, is a simpler and older resampling tool, and permutation tests shuffle the group labels to build a null distribution for a test statistic under the hypothesis of no group difference. These methods are especially valuable for statistics whose sampling distribution is hard to derive analytically, such as a median, a capability index, or a complex model coefficient. When the theory-based formulas are in doubt, the bootstrap provides a trustworthy, assumption-light answer. **Sample size and power analysis determine how many observations are needed to detect an effect of a given size.** The power of a test is the probability of rejecting the null when a real effect exists, and it grows with the sample size, the effect size, and the significance level, so an experiment must be sized to have adequate power for the effect the engineer cares about. The required sample size for comparing two means depends on the target difference, the process variability, the significance level, and the desired power, and the standard formula shows that the sample size grows as the variance divided by the square of the target difference. An underpowered study risks a type II error and may falsely conclude there is no effect, while an overpowered study wastes resources detecting trivial differences. Before running a comparison or a designed experiment, an engineer should size the study to give the analysis a real chance to answer the question. **Outliers, missing data, and data quality determine whether the analysis can be trusted at all.** An outlier is an observation that is far from the rest of the data, and it can be a genuine rare event, a measurement error, or a data-entry mistake, so it must be investigated rather than mechanically deleted, with robust summaries such as the median and interquartile range resisting its influence. Missing data arise from failed measurements, dropped wafers, or censoring, and how they are handled, by deletion or by imputation, depends on whether they are missing at random, with careless handling biasing the results. The first duty of any analysis is to check the data for plausibility, range, duplicates, and coding errors, because a clean analysis of dirty data produces confident but false conclusions. In the fab, a single mis-keyed thickness or a failed probe measurement can flip a yield conclusion if it is not caught early. | Statistical Test | Data Type | Question Answered | Assumptions | |---|---|---|---| | One-sample t-test | continuous, 1 group | mean differs from target? | approx. normal, independent | | Two-sample t-test | continuous, 2 groups | means differ? | approx. normal, independent | | Paired t-test | continuous, paired | paired difference nonzero? | diff approx. normal | | One-way ANOVA | continuous, 3+ groups | any group mean differs? | normal, equal variance | | Chi-square test | categorical counts | counts match expected? | expected counts not too small | | Pearson correlation | 2 continuous | linear association? | linear, no strong outliers | | Linear regression | response + predictors | effect of predictors on response? | linear, independent errors | | Mann-Whitney U | continuous, 2 groups | distributions differ? (rank) | independent, ordinal | | Kruskal-Wallis | continuous, 3+ groups | distributions differ? (rank) | independent, ordinal | | Bootstrap | any | confidence interval? | sample representative | **Statistical thinking is a way of reasoning about evidence, variation, and risk, not a bag of formulas.** A data-informed engineer knows that every measurement carries error, that every estimate carries uncertainty, and that every comparison is subject to error, and this mindset prevents the classic mistakes of over-trusting a p-value, over-plotting an outlier, and over-extrapolating a model. The reproducibility crisis across science has sharpened the demands on statistics: preregistration, multiple-testing control, effect sizes, confidence intervals, and validation on held-out data are now expected practice rather than optional refinements. When a machine-learning model is fit to fab data, the same foundations apply, with training-test splits, cross-validation, and calibration guarding against overfitting and overconfidence. The goal of statistics basics is not to memorize tests but to develop the judgment to choose the right method, interpret its output honestly, and communicate the uncertainty to the decision maker. ```flowchart A[Research / engineering question] --> B[Choose target and hypotheses] B --> C[Design and size the study] C --> D[Collect data via random sampling] D --> E[Describe: mean, spread, plots] E --> F[Check assumptions and data quality] F --> G{Assumptions met?} G -->|Yes| H[Parametric test or regression] G -->|No| I[Nonparametric / rank or bootstrap] H --> J[Compute statistic, p-value, CI] I --> J J --> K{Significant?} K -->|No| L[Report no detectable effect, with CI] K -->|Yes| M[Estimate effect size & confidence] M --> N[Check for multiple-testing / confounding] N --> O[Make a defensible engineering decision] L --> O O --> P[Document assumptions & uncertainty] ``` | Measure | Definition | Robustness | Typical Use | |---|---|---|---| | Mean | sum of values ÷ n | sensitive to outliers | symmetric data, capability | | Median | middle value when sorted | robust to outliers | skewed data, lifetime | | Mode | most frequent value | robust | categorical / peaks | | Range | max − min | very sensitive | quick spread check | | Interquartile range | Q3 − Q1 | robust | box plots, skewed data | | Variance | avg squared deviation | sensitive | ANOVA, t-tests | | Standard deviation | √variance | sensitive | capability, process spread | **The division of statistics into descriptive and inferential branches gives an engineer a complete toolkit from exploration to decision.** Descriptive statistics summarize what has been observed, with means, medians, variances, histograms, and box plots, and inferential statistics draw conclusions about what has not been observed, using sampling, confidence intervals, tests, and models to reach beyond the sample. The flow from collecting data to describing it, to estimating parameters, to testing hypotheses, to building models, and finally to deciding, is the arc that every real analysis follows, and each stage has its own tools and its own pitfalls. A semiconductor engineer who masters this arc can turn the flood of metrology, test, and design data into a clear, quantified, defensible answer to the question of whether a process is stable, whether a change helped, and whether a design can be trusted. Read statistics basics through a decision-and-evidence lens rather than a formula-memorization lens.