Statistics

Statistics studies how to learn from data while accounting for variation and uncertainty. Descriptive statistics summarizes observations; statistical inference uses an explicit statistical model to estimate unknown quantities or assess claims. For example, the sample mean estimates a population mean, while its standard error describes variation across hypothetical repeated statistical samples. A large dataset does not automatically remove bias from statistical sampling or confounding. The design generating the observations is part of the reasoning, rather than an optional prelude to calculating a formula.

Computational statistics develops and studies computational procedures for statistical inference. It includes numerical evaluation of likelihood functions, simulation from probability distributions and assessment of uncertainty when an exact symbolic calculation is unavailable. The statistical question and the numerical approximation both need error control.

For example, if independent simulations $X_1,\dots ,X_M$ follow a model and $g(X)$ has finite variance, the Monte Carlo estimator $\hat{\mu }=M^{-1}{\sum }_{j}g(X_j)$ estimates $\mu =\mathbb{E}[g(X)]$, with variance $\mathrm{Var}(g(X))/M$. Halving its standard deviation therefore requires four times as many independent simulations. An optimization routine finding a maximum of a likelihood function and a simulation routine estimating an integral solve different computational tasks, even when both are steps in the same analysis.

Econometrics uses statistical models to quantify economic relationships and to assess predictions or causal claims from economic data. The choice of variables and the mechanism generating observations matter as much as estimating the model coefficients.

For example, in a linear regression of log earnings on years of schooling, a slope $0.08$ corresponds to a fitted earnings ratio $e^{0.08}\approx 1.083$ for one additional year. Interpreting that ratio as a causal effect needs more than a good fit: prior attainment or family circumstances may affect both schooling and earnings, creating confounding. Randomization, suitable instrumental variables or a credible natural experiment can support causal interpretation under additional assumptions. The numerical regression coefficient alone supplies an association within its model, not those assumptions.

Mathematical statistics studies the mathematical properties of procedures for learning from data: estimation, statistical hypothesis tests and quantification of uncertainty under explicit statistical models. It asks what a method estimates, how its error behaves and which assumptions justify that behavior.

If $X_1,\dots ,X_n$ are independent and identically distributed random variables with expected value $\mu$ and finite variance ${\sigma }^2$, the sample mean $\overline{X}$ satisfies $\mathbb{E}\overline{X}=\mu$ and $\mathrm{Var}(\overline{X})={\sigma }^2/n$. Independence removes cross-covariances in the second calculation. These identities explain why averaging can improve precision, while also exposing a limitation: strongly dependent observations may provide much less information. Theorems about estimators are therefore conditional statements about a sampling model, rather than universal guarantees for every dataset.

Biostatistics develops and applies statistical inference to biological and health questions. Its problems include designing experiments, comparing treatments, measuring biological variation and analyzing survival or repeated observations. The experimental unit must match the independent unit of information: multiple measurements of one organism need not substitute for measurements of multiple organisms.

Suppose a randomized study observes an event in 10 of 100 treated participants and 20 of 100 controls. The estimated risk difference is $0.10-0.20=-0.10$, while the estimated relative risk is $0.10/0.20=0.5$. These express different scales of comparison, and neither alone expresses its uncertainty. Assignment, follow-up and the definition of the event determine what effect can be estimated. For incomplete follow-up, methods from survival analysis distinguish observing no event yet from establishing that an event will never occur.

Geometric data analysis represents multivariate observations as weighted clouds of points, then studies their Euclidean geometry through distances, orthogonal projections and principal components. Principal component analysis applies this approach to numerical variables; correspondence analysis adapts it to a table of counts with its own weights and metric. Choosing the weights and metric is part of the analysis, since these choices decide which differences between observations count as large.^[ref]

For a column-centered data matrix with singular value decomposition $X=U\Sigma V^T$, the first $k$ principal components give coordinates $X{V}_k=U_k{\Sigma }_k$. They are the orthogonal projection onto the chosen principal directions. Rescaling one variable changes the distances and can change those directions: measurements in metres and millimetres should not receive different influence merely because the units were changed. The geometry must therefore be interpreted together with the scaling convention, rather than treating the resulting plot as an intrinsic picture of the observations.

Correspondence analysis represents a nonnegative contingency table geometrically, displaying how row and column profiles depart from independence. Let $P$ be the table divided by its grand total, and let $r=P1$, $c=P^{T}1$ be positive marginal proportions. Its standardized residual matrix isA singular value decomposition of $S$ provides principal axes; the sum of squared singular values equals the Pearson Pearson chi-squared statistic for contingency tables divided by the grand total. Thus weighted departures from independence become geometric variation.^[ref]

For counts $\left(\begin{array}{cc}30& 10\\ 10& 30\end{array}\right)$, the total is 80 and both margins are $(1/2,1/2)$. Then $S=\left(\begin{array}{cc}1/4& -1/4\\ -1/4& 1/4\end{array}\right)$ has one nonzero singular value $1/2$. The total inertia is therefore $1/4$ and the Pearson chi-squared statistic for contingency tables is 20. The association has one geometric dimension. Zero margins require removal or separate treatment before using the inverse square roots.

A randomized mechanism $M$ satisfies differential privacy with parameters $(\epsilon ,\delta )$ if, for every ordered pair of adjacent datasets $D,D^{\prime }$ and measurable output event $S$,Adjacency specifies the contribution of one individual, such as adding or removing one record. The probability is over the mechanism's randomness.

For an integer count query with sensitivity one under add/remove adjacency, releasing the count plus independent Laplace distribution noise of scale $1/\epsilon$ gives $(\epsilon ,0)$ privacy. The ratio of output densities is at most $e^{\epsilon }$ by the triangle inequality. The guarantee limits distinguishability between neighboring inputs; it does not require publishing identical results or certify that every statistical estimate is accurate.^[ref]

The Laplace mechanism releases a numerical query with added Laplace distribution noise scaled to the query's sensitivity. For $f(D)\in {\mathbb{R}}^k$, define its sensitivity under the chosen dataset adjacency byFor $\epsilon >0$ and positive finite sensitivity, release $f(D)+(Y_1,\dots ,Y_k)$ with independent noise coordinates of density $p(y)=(2b{)}^{-1}{e}^{-|y|/b}$, where $b={\Delta }_{1}f/\epsilon$. The ratio of output densities for adjacent inputs is bounded by $\exp (\Vert f(D)-f(D^{\prime }){\Vert }_1/b)\le e^{\epsilon }$, by the triangle inequality. Integrating over any measurable output event proves $(\epsilon ,0)$ differential privacy.^[ref]

For a count with add/remove sensitivity one and $\epsilon =0.5$, the noise scale is two and its variance is $2b^2=8$. Smaller privacy parameters increase uncertainty in this mechanism. A fixed output when sensitivity is zero needs no noise. The adjacency convention is essential: changing what one person can contribute changes the required sensitivity bound.

High-dimensional statistics studies inference when the number of variables or parameters is large relative to the number of observations, often allowing both to grow. The difficulty is structural rather than merely computational: with an $n\times p$ design matrix and $p>n$, the null space is nontrivial, so $X\beta =X(\beta +v)$ for every $v$ in that null space.

The data alone then cannot identify an arbitrary coefficient vector. Assuming a sparse vector of regression coefficients can make restricted recovery possible when the design matrix is injective on the allowed differences. Regularization encodes or exploits such structural assumptions. A fitted model with many parameters is therefore not automatically informative about every parameter; identifiability and uncertainty depend on the model's structure and sampling design.

A statistical population is the collection of units, outcomes or distributional cases about which an analysis seeks conclusions. A finite population may consist of $N$ fixed values $y_1,\dots ,y_N$, with population mean ${\mu }_N=N^{-1}{\sum }_i{y}_i$. A superpopulation model instead describes observations through a probability distribution.

These viewpoints support different sources of randomness. In sampling from a fixed finite population, selection is random even if each value is fixed. Under simple random sampling without replacement, every unit has inclusion probability $n/N$, and the sample mean is unbiased for ${\mu }_N$. The accessible sampling frame can differ from the target population, producing a coverage problem that a large sample alone cannot repair.

A probability plot compares order statistics with plotting positions or model quantiles to diagnose whether a proposed probability distribution describes data. If the model can be transformed into a linear identity $T(F(x))=a+bS(x)$, plot $T(i/(n+1))$ against $S(X_{(i)})$. Approximate linearity supports that model's shape; systematic curvature suggests a mismatch.

The plotting position $i/(n+1)$ comes from the exact mean of the $i$th uniform order statistic, by the probability integral transform. It does not assert equality between the expectation of a nonlinear transformation and that transformation of the expectation. Fitted slope and intercept can estimate model parameters, but the visual plot alone is not a calibrated hypothesis test.

For a sample or finite statistical population of positive size $n$, the relative frequency of a category represented $n_i$ times is $n_i/n$. It is a proportion and differs from an oscillation frequency.

Demography measures the size, composition and change of human populations. It combines population censuses, registrations of births and deaths, and statistical sampling with mathematical models of population dynamics. A population census refers its counts to a specified time and territory; collecting questionnaires on different days should not silently change the reference date.^[ref]

For a fixed territory, a demographic accounting identity iswhere $B,D,I,E$ count births, deaths, immigration and emigration during the intervening period. For example, $N(t_0)=10000$, $B=140$, $D=90$, $I=80$ and $E=110$ give $N(t_1)=10020$. This demographic balancing equation distinguishes natural increase from net human migration; it does not assume an exponential growth law.

Counts also need denominators. A birth rate compares births with the relevant population exposure, rather than comparing raw birth totals between differently sized human populations. Changes in age composition can alter an aggregate rate even when the underlying age-specific rates remain fixed, motivating analysis by demographic cohorts.

For demography, human migration is movement that changes a person's place of usual residence across a specified territorial boundary. Counts depend on the boundary, the residence criterion and the observation interval. A commuter crossing daily and a person changing residence therefore need not be counted alike.

In the demographic balancing equation, immigration $I$ adds residents and emigration $E$ removes them, so net migration is $I-E$. If a district gains 800 and loses 500 residents by migration during a year, its net migration is 300, while the total number of moves is 1300. Net migration alone can conceal substantial turnover. Movement between two districts contributes to each district's account but cancels in the national total if both districts are inside the same national boundary. This accounting viewpoint is the quantitative aspect used here.

A demographic cohort is a group defined by experiencing a specified event during the same interval, then followed through time. A birth cohort consists of people born in a stated period; other examples group people by year of immigration or first diagnosis. It is not the same as all people of a given age in a single cross-section.

Ignoring migration, a birth cohort with 1000 initial members and 950 survivors ten years later has ten-year survival proportion $950/1000=0.95$. Comparing that with the proportion of today's ten-year-olds among today's residents answers a different question, because births, deaths and human migration have affected the denominator. Cohort analysis separates changes experienced by a group as it ages from changes between successive groups. For a cohort defined by birth year $c$, calendar year and age satisfy $t=c+a$, which creates an exact dependence in attempts to estimate age, period and cohort effects simultaneously.

For a fixed territory and interval, the demographic balancing equation is the accounting identitywhere $N_0,N_1$ are initial and final resident counts and $B,D,I,E$ are births, deaths, immigration and emigration during that interval. All quantities must use compatible definitions of territory, residence and reference time. This is conservation of population counts, not an assumed law of population dynamics.

For $N_0=10000$, $B=140$, $D=90$, $I=80$ and $E=110$, the final count is $10020$: natural increase adds 50 and net human migration removes 30. Rearranging yields $I-E=N_1-N_0-B+D$, allowing net migration to be estimated residually. In measured data, that residual also absorbs errors in censuses and vital-event counts, so the identity does not independently establish the accuracy of any measurement.

A population census is the organized enumeration or equivalent statistical accounting of the people in a specified territory at a specified reference time, including selected demographic characteristics. Universality and a common reference time distinguish the target of the count from the days on which information happens to be collected.^[ref]

For example, if interviews span two weeks, a person born after census night should not silently become part of the census-night population merely because the household is interviewed later. Rules about usual residence also prevent a student or traveler from being counted inconsistently across locations. The resulting counts supply denominators for rates and benchmarks for statistical sampling. Attempting complete coverage does not eliminate omission, duplication or classification errors; a census is a measurement procedure, not a guarantee of perfect counts.

A human population for quantitative demography is the collection of people satisfying a stated territorial, residence and time definition. Its size $N(t)$ is a count; its age or other composition describes how that count is distributed between subgroups. A resident population and the people physically present on a particular night can have different sizes.

For example, a town with 10000 residents and 200 births during one year has an approximate crude birth rate of 20 births per 1000 residents per year if its midyear population is 10000. A town with 20000 residents and the same birth count has approximately half that rate. Raw event counts therefore cannot replace denominators. Population censuses and the demographic balancing equation connect successive measurements, while demographic cohorts allow changes in composition to be tracked.

A demographer studies human populations quantitatively: how many people are present, how their characteristics are distributed, and how births, deaths and human migration change those counts. The work connects population census definitions and statistical sampling to statistical estimation and mathematical models of population dynamics. A forecast must distinguish observed counts from assumptions about future rates.^[ref]

For example, in a closed demographic cohort with $n_a(t)$ people aged $a$ and common one-year survival probability $s_a(t)$ conditional on that count, a basic projection isA cohort of 1000 people with survival probability $0.998$ contributes an expected 998 survivors to the next age class. Births create a new youngest class, and human migration requires additional terms. This elementary calculation illustrates why a demographer works with age structure and explicit statistical parameters, rather than extrapolating a single total without assumptions.

Thomas Robert Malthus made the comparison between population growth and the growth of food supply explicit in his 1798 An Essay on the Principle of Population by Thomas Robert Malthus. In Chapters I–II, he assumed that unchecked numbers could follow a geometric progression, taking doubling every 25 years as his rule, while food supply gained a fixed amount in each such period, an arithmetic progression. These were different hypotheses about two quantities, rather than a calculation from a single universal growth law.^[ref]

With equal initial scales, his comparison can be written as $N_k=N_0{2}^k$ and $F_k=F_0(1+k)$ after $k$ periods of 25 years. Then $N_k/F_k=(N_0/F_0)2^k/(1+k)$ grows without bound. The corresponding continuous exponential growth rate for numbers would be $\log (2)/25$ per year. The mathematics describes the consequence of the assumed rates; it does not establish that actual birth, death, human migration and food-production rates stay fixed. Demography can test population assumptions against counts, age composition and measured rates.