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A prediction interval is a random interval designed to contain a future observation with prescribed repeated-sampling probability. Unlike a confidence interval for a mean response, it includes both uncertainty in the fitted mean and the new observation's irreducible random variation.

For independent observations uniform on $(\theta -1/2,\theta +1/2)$, let $U$ and $V$ be the sample minimum and maximum, $R=V-U$, and $C=(U+V)/2$. For $n\ge 2$ their joint density in $(c,r)$ is $n(n-1)r^{n-2}$ on $0<r<1$, $|c-\theta |<(1-r)/2$. Integrating over $c$ gives $f_R(r)=n(n-1)r^{n-2}(1-r)$, independent of $\theta$, so $R$ is an ancillary statistic. Moreover $C\mid R=r$ is uniform on $(\theta -(1-r)/2,\theta +(1-r)/2)$. Consequentlyis an exact conditional $(1-\alpha )$ confidence interval for $\theta$. The realized range determines the appropriate precision, although the range is not itself evidence about the value of $\theta$.

Suppose $0<p_L\le p\le p_U<1$ and $0<a_L\le a\le a_U<1$. The formula for relative risk from group compositions increases with $a$ and decreases with $p$, since $a/(1-a)$ increases and $(1-p)/p$ decreases. This monotonicity proves the displayed bounds. If percentages are rounded to the nearest whole percentage point, each corresponding proportion lies within $0.005$ of its displayed value. These bounds describe rounding uncertainty; they are not a confidence interval for sampling uncertainty.

A return level plot graphs $T$ horizontally and $z_T$ vertically, often using a logarithmic scale for $T$. To read it, choose a return period, move up to the curve, and read the corresponding return level. Conversely, a threshold's horizontal coordinate gives the reciprocal of its annual exceedance probability. The usual plot of fitted quantiles against return periods, and optional pointwise confidence intervals, are documented in Mathieu Ribatet, POT package: return level plot.

The near-straight shape has a mathematical explanation: since $-\log (1-1/T)\sim 1/T$ as $T\to \infty$, the Gumbel distribution gives $z_T=\mu +b\log T+o(1)$. Multiplying a large return period by ten therefore increases the return level by approximately $b\log 10$, about $4.605$ units for this toy curve. A curve indexed by rarity puts several thresholds on one common scale while retaining their measurement units.

For a fitted curve, a pointwise confidence interval concerns uncertainty in the estimated quantile at a specified $T$; it is distinct from a prediction interval for a future annual sample maximum, and pointwise intervals need not form a simultaneous confidence band. A far-right curve value can also be an extrapolation beyond the observation record: extending the line adds no observations to support the chosen tail model. The diagram above uses fixed toy parameters, so it shows neither estimation uncertainty nor an observational record. Its interpretation throughout assumes the stated annual block and stationarity.

The 1975 experiment mixed powdered sodium chloride and 0.5%-chromium ruby in a 4:1 volume ratio. An Inconel gasket, initially $0.15\mathrm{mm}$ thick with a $0.3\mathrm{mm}$ hole, held the powder and 4:1 methanol–ethanol. At $25{}^{\circ }\mathrm{C}$, X-ray diffraction rings on film $12\mathrm{cm}$ away gave salt compression; a reference equation of state assigned pressure. After decreasing load, they waited at least five hours and bracketed diffraction exposures with $R_1$ readings to check drift.^[ref]

The recommended linear slope was $2.740\mathrm{GPa}\mathrm{n}{\mathrm{m}}^{-1}$, with a $95\%$ fit confidence interval of $\pm 0.016\mathrm{GPa}\mathrm{n}{\mathrm{m}}^{-1}$, to $19.5\mathrm{GPa}$. Reference-scale systematic measurement uncertainty remained separate. Above about $10.4\mathrm{GPa}$ the medium was only quasihydrostatic; separate range fits checked consistency.^[ref]

The logical chain is compression to reference pressure, paired with optical shift, then a fitted response. X-ray diffraction supplies the reference comparison during calibration; subsequent ruby readings transfer that scale to other samples. The reference equation of state is therefore part of the meaning of the reported pressure. Changing it can move all calibrated readings together, even if the optical measurements repeat perfectly. A regression residual tests agreement with the chosen reference within the sampled conditions; it cannot independently certify that reference.

Fix $N>1$ real population values $y_1,\dots ,y_N$. Define the finite population mean and two normalizations of the finite population variance byThe values are fixed; only the selected subset $A$ is random. For its sample mean ${\bar{y}}_A=n^{-1}{\sum }_{i\in A}{y}_i$, the exact design expectation and variance areThe sample mean is an unbiased estimator of the finite population mean. At $n=N$, every unit is observed and ${\bar{y}}_A=\bar{Y}$ exactly, so the variance is zero. The $N=1$ case is deterministic and is handled directly, since $S^2$ is undefined then.^[ref]

The finite population correction depends on the baseline. With the $N-1$ normalization it is $1-n/N$ multiplying $S^2/n$. With the $N$ normalization it is $(N-n)/(N-1)$ multiplying ${\sigma }_N^2/n$, the exact variance for independent uniform sampling with replacement. The standard error uses the square root of the relevant variance factor. Mixing the two normalizations gives a wrong answer even at $n=1$, when replacement cannot matter. For the six-value example, $\bar{Y}=7$, ${\sum }_i(y_i-7{)}^2=70$, $S^2=14$ and ${\sigma }_N^2=35/3$, reproducing $14/3$ without replacement and $35/6$ with replacement.

A short proof exposes the statistical dependence. Let $X_r$ be the value at position $r$ in uniform sequential sampling without replacement, so each $X_r$ has expected value $\bar{Y}$ and variance ${\sigma }_N^2$. Write $d_i=y_i-\bar{Y}$. Every ordered pair of distinct labels is equally likely, and ${\sum }_i{d}_i=0$, hence for $r\ne s$,Linearity of expectation proves unbiasedness without requiring independence. The variance of a sum includes all $n(n-1)$ ordered covariance terms, givingFor the original population, each pair of distinct draw positions has covariance $-7/3$. Drawing a high value leaves a lower mean among the remaining values, balancing subsequent draws. Equal marginal probability distributions therefore coexist with negative covariance.

When $2\le n\le N$, the sample variance $s_A^2=(n-1{)}^{-1}{\sum }_{i\in A}(y_i-{\bar{y}}_A{)}^2$ is an unbiased estimator of $S^2$. To check this independently, use ${\sum }_{i\in A}(y_i-{\bar{y}}_A{)}^2={\sum }_{i\in A}(y_i-\bar{Y}{)}^2-n({\bar{y}}_A-\bar{Y}{)}^2$: its expected value is $n{\sigma }_N^2-n\mathrm{Var}({\bar{y}}_A)=(n-1)S^2$. Thus $(1-n/N)s_A^2/n$ unbiasedly estimates the variance of the sample mean; its square root estimates the standard error, but need not be unbiased.^[ref]

The diagram compares variance with the expected fraction of distinct population labels observed. Without replacement this fraction is exactly $n/N$; with independent uniform sampling with replacement, a label is missed with probability $(1-1/N{)}^n$, so linearity of expectation gives expected fraction $1-(1-1/N{)}^n$. At six draws, replacement covers only about $66.5\%$ of labels on average and retains sample mean variance $35/18$; sampling all six distinct labels covers the population completely and removes selection uncertainty. This population coverage is distinct from confidence interval coverage probability.

The correction matters when the sampling fraction is appreciable. It calibrates uncertainty about this fixed population under the specified selection mechanism. It does not correct unequal selection weights, missing observations or measurement errors; observing every label also does not eliminate uncertainty about a larger unobserved population. Moreover, an exact variance alone does not guarantee the coverage probability of a proposed confidence interval: a small discrete sampling distribution can make a normal approximation inaccurate.

Vary the unobserved outcome probabilities for item and unit nonresponders, and any unknown auxiliary-variable mixture, over scientifically plausible ranges. Recompute the population estimate and identify values at which substantive conclusions change. This probes untestable transport or missing at random assumptions; it is distinct from sampling uncertainty represented by a confidence interval. Letting each missing-outcome probability vary from zero to one gives worst-case identification bounds.

A simple random walk on a locally finite graph chooses each vertex in the current vertex's graph neighbourhood with equal probability, assuming there is at least one available neighbour. On the integer line there are two choices, one unit left or right, so each has probability $1/2$. This is the one-dimensional simple symmetric random walk. Terminology varies: a simple random walk on the integer line is also often allowed unequal step probabilities, with “symmetric” specifying the fair case.

For a simple symmetric random walk starting at $S_0=0$, each increment has the Rademacher distribution, with expected value zero and variance one. Thus linearity of expectation and the variance of a sum of independent random variables giveFor example, after $80$ steps the expected value is zero, the variance is $80$, and the standard deviation is $\sqrt{80}\approx 8.944$. Doubling the typical distance scale takes four times as many steps. This square-root growth explains why a random walk can wander far even when there is no preferred direction.

The colored curves are illustrative sample paths, not averages over all possible walks. The dashed curves show the expected values; the shaded bands show the expected value plus or minus one standard deviation at each time. These bands are a scale of spread, not confidence intervals, and give no guarantee that an entire sample path stays inside them.

Zero expected value does not make this a stationary process: its one-time probability distribution changes with $n$, as the increasing variance already proves. It does have stationary increments and independent increments, because blocks of fresh steps have the same probability distribution at any starting time and disjoint blocks use independent random variables. This separation of position from increment is one reason the random walk is such a useful first example of a stochastic process.

Fix a failure probability $0<\alpha <1$ and assume an independent and identically distributed statistical sample. A finite-sample confidence band for its population cumulative distribution function isHere $\log$ is the natural logarithm. The DKW inequality impliesThis is simultaneous coverage over the whole real line, valid at each fixed sample size.^[ref]

To obtain the band, solve $2e^{-2n{\epsilon }^2}=\alpha$ for the positive $\epsilon$. Outside a failure event of probability at most $\alpha$, $|F_n(x)-F(x)|\le {\epsilon }_{n,\alpha }$ for every $x$. Since a cumulative distribution function already lies in $[0,1]$, clipping the endpoints to this interval preserves the coverage event. The guarantee is at least $1-\alpha$, and need not equal it.

Choose $\alpha =0.1$, giving at least $90\%$ simultaneous coverage under the stated model. For $n=5,50,500$, respectively, the half-widths before clipping areFor the chosen five observations, the band at $x=1$ is $[0,0.94733]$, while at $x=4$ it is approximately $[0.25267,1]$. At $x=3$ the unclipped band would be $[0.05267,1.14733]$, so clipping gives $[0.05267,1]$. The broad uncertainty is informative: five observations give a precise description of themselves, but a coarse bound for an unknown population cumulative distribution function.

A collection of $90\%$ pointwise confidence intervals only promises coverage separately at each preselected threshold; it does not promise $90\%$ coverage of an entire cumulative distribution function. Here the one simultaneous event permits inspecting any threshold after seeing the data. The probability refers to repeated sampling of the entire statistical sample, with $F$ fixed. It does not assign $90\%$ probability to a fixed $F$ after observing this particular band, and it is not a prediction interval for a future observation. The displayed $n=5,50,500$ comparison gives a guarantee for each fixed $n$ separately, not simultaneous validity over all sample sizes or an optional stopping rule. The chosen observations are an illustration; the coverage claim follows from the DKW inequality, not from their appearance or a simulation. Dependence or observations with different sampling laws require another argument.

At fixed $\alpha$, multiplying $n$ by $100$ divides the half-width by $10$. Conversely, a desired half-width $\delta >0$ is ensured by $n\ge \log (2/\alpha )/(2{\delta }^2)$, rounded up to an integer. For $\alpha =0.1$ and $\delta =0.1$, this sufficient sample size is $150$. This makes the confidence band useful for planning how many observations are needed before estimating threshold probabilities to a chosen simultaneous accuracy.

Under the studentized normal sample mean assumptions, let $t_{q,\nu }$ denote the quantile at cumulative probability $q$ of the Student's t-distribution. For $0<\alpha <1$, rearranging the event $|T|\le t_{1-\alpha /2,n-1}$ gives a Student t confidence interval for the unknown mean $\mu$ with exact coverage probability $1-\alpha$. ^[ref]

Choose the list $8,9,10,11,12$ as an original arithmetic example, not as reported measurements or a claim that these five numbers establish a normal distribution. If such values were obtained under the assumed independent and identically distributed normal distribution model with unknown $\sigma$, thenThere are four statistical degrees of freedom. For a 95% confidence interval, $t_{0.975,4}=2.776445105\dots$, soUsing the standard normal quantile $z_{0.975}=1.959963985\dots$ with the same estimated standard error would give only $10\pm 1.385903824\dots$. That underestimates the correct margin by about 29.4%. Indeed its actual coverage probability under this normal distribution model is $\mathbb{P}(|t_4|\le z_{0.975})\simeq 0.8784$, rather than 0.95.

The observations, the sample mean and the confidence interval all have different meanings. The bar estimates the population mean; it is not an interval intended to contain 95% of individual observations. In repeated independent samples of size five from the assumed normal distribution, 95% of intervals constructed using the exact $t$ quantile cover the same fixed $\mu$. This is not a 95% posterior probability that the fixed statistical parameter lies in this particular realized interval. Such a probability requires a separate Bayesian inference model.

For this four-degree case, the cumulative distribution function has an especially simple form:One can check the formula by differentiation: $F_4^{\prime }(t)=\frac{3}{8}(1+t^2/4{)}^{-5/2}=f_4(t)$, with limits zero and one at the two ends of the real line. Solving the increasing equation $F_4(t)=0.975$ by the bisection method gives the quantile used above.

Statistical estimation uses observed data to infer numerical properties of a population or model. A point estimator gives a value, while a confidence interval describes uncertainty with a specified coverage property. Maximum likelihood estimation is one approach; bias, variance, identifiability and the sampling or selection mechanism all affect an estimate's interpretation.

Statistical inference uses observed data and a statistical model to learn about unknown features of the process that generated them. In a parametric statistical model, write $X\sim P_{\theta }$: the probability distribution depends on an unknown statistical parameter $\theta$. An estimand specifies the target, such as a mean or a difference of means. A statistic is a function of the observations; an estimator is a statistic used to estimate the target. Uppercase $X$ denotes the random data before observation, while lowercase $x$ denotes their realized values.

Three questions lead to different outputs. Statistical estimation asks which target value the observations suggest. A confidence interval asks for a range produced by a procedure with calibrated coverage probability. A hypothesis test asks whether the observations depart sufficiently from a specified null hypothesis to enter its rejection region. These outputs complement one another: an estimated effect size describes magnitude, while a standard error describes the sampling distribution of an estimator. A p-value describes a tail probability under the null hypothesis.

The appeal of statistical inference is that it turns variable observations into reproducible uncertainty statements. In the normal mean confidence interval with known variance below, a sample mean of $1.2$ has standard error $0.5$: both the estimated shift and its precision matter. The calibration depends on the statistical model, including independence and the specified probability distribution. Increasing sample size improves precision under those assumptions; it does not repair a systematically misleading sampling procedure. Bayesian inference provides another framework, using a prior distribution and a likelihood function to obtain a posterior distribution.

For two independent observations uniform on $[\theta -a,\theta +a]$, the displayed interval of possible parameter values contains the true parameter almost surely. Intersecting any confidence interval with it preserves that interval's coverage, while excluding values inconsistent with the observations. The order statistics interval $[U_{(1)},U_{(2)}]$ has coverage $1/2$, since the observations straddle $\theta$ with probability $1/2$. If their separation exceeds $a$, the entire interval of possible parameters lies between the observations; this explains why the nominal one-half procedure can be unnecessarily wide on that event.