1 General concept

1.1 Definition and interpretation

In statistics, discrepancy describes the extent to which two quantities fail to agree. It is most often used to compare observed data with expected, predicted, or reference values. A small discrepancy suggests close correspondence, while a large one indicates that the data depart noticeably from what was anticipated.

The term is broad rather than technical in a single fixed sense. In one setting it may refer to a difference between measured and true values; in another, to a statistic used to judge whether a model fits the data well. Because of this flexibility, the interpretation of a discrepancy depends strongly on context.

1.2 Observed versus expected values

A common use of discrepancy arises when observed values are compared with expected values. For example, in a survey or experiment, the observed counts in categories may be compared with counts predicted by a theory or model. The resulting difference is the discrepancy, and its size helps indicate whether the observed pattern is consistent with expectation.

This comparison can involve raw data, percentages, probabilities, or fitted values from a statistical model. In many applications, the discrepancy is not interpreted by itself but as part of a larger evaluation of fit, uncertainty, or reliability.

1.3 Sources of discrepancy

Discrepancy can arise from several causes. Some reflect random variation, while others point to systematic problems in measurement, sampling, or modeling. Distinguishing among these sources is important because different causes require different responses.

1.3.1 Measurement error

Measurement error occurs when recorded values differ from the quantities they are intended to represent. This may result from imprecise instruments, human recording mistakes, or unstable measuring conditions. Such errors create discrepancies between the observed data and the underlying values of interest.

1.3.2 Sampling variation

Even when measurements are accurate, random sampling can produce differences between a sample and the population it represents. These discrepancies are expected to some degree and are often quantified through standard errors, confidence intervals, or test statistics.

1.3.3 Model misspecification

A model may be unable to capture the structure of the data. Missing variables, incorrect functional forms, or unrealistic assumptions can all generate systematic discrepancies between model predictions and observations. In such cases, the discrepancy may reveal that the model is inadequate rather than that the data are unusual.

2 Discrepancy measures

2.1 Absolute and relative differences

The simplest discrepancy measures are based on subtraction or division. An absolute difference is the magnitude of the gap between two values, ignoring direction. A relative difference expresses the gap in proportion to a reference value, which can be useful when quantities are measured on very different scales.

These measures are easy to interpret, but they may not capture the full structure of multivariate data or correlated errors. For that reason, more specialized measures are often used in statistical analysis.

2.2 Residual-based measures

Residuals are among the most common discrepancy measures in statistics. A residual is typically defined as the observed value minus the fitted or expected value. Large residuals indicate that the model does not closely match the data at particular points.

Residual-based measures are central in regression, time series analysis, and other modeling frameworks. They help identify lack of fit, unusual observations, and patterns that may suggest violated assumptions.

2.3 Distance-based measures

Distance-based discrepancy measures summarize how far observations lie from expected values in a geometric or metric sense. These measures are useful when multiple variables are involved and simple one-dimensional differences are not sufficient.

2.3.1 Euclidean discrepancy

Euclidean discrepancy treats the difference between two points as a straight-line distance in coordinate space. It is commonly used when variables have been standardized or are otherwise comparable. This measure is intuitive, but it may not account for correlations among variables.

2.3.2 Mahalanobis discrepancy

Mahalanobis discrepancy incorporates the covariance structure of the data. By taking account of scale and correlation, it measures how unusual a point is relative to a multivariate distribution. It is often used for identifying multivariate outliers and for comparing observations in correlated data sets.

2.4 Aggregate discrepancy statistics

In many settings, individual differences are combined into a single summary statistic. Examples include sums of squared residuals, chi-square statistics, and other goodness-of-fit measures. Such aggregates make it easier to compare models or test hypotheses, though they may obscure the location of particular mismatches.

These statistics often serve as the basis for formal inferential procedures. Their usefulness depends on the assumptions underlying the model and the distribution of the data.

3 Discrepancy in statistical inference

3.1 Goodness-of-fit testing

Goodness-of-fit methods assess whether observed data are consistent with a specified distribution or model. The discrepancy between observed and expected frequencies or values is converted into a test statistic, which is then compared with a reference distribution or simulated benchmark.

A small discrepancy supports the idea that the model fits reasonably well, while a large one may suggest poor fit. However, the practical meaning of a discrepancy depends on sample size, variability, and the complexity of the model.

3.2 Hypothesis testing

In hypothesis testing, discrepancy helps quantify how far the observed data are from what would be expected if a null hypothesis were true. Test statistics are designed to increase when the data depart more strongly from the null model.

The resulting p-value or decision rule does not measure discrepancy directly, but it uses discrepancy as the basis for inference. In this way, the concept links observed evidence to formal statistical judgment.

3.3 Model diagnostics

Model diagnostics examine whether a fitted model adequately represents the data. Discrepancy plays a central role because diagnostic tools look for systematic departures between fitted values and observations.

3.3.1 Residual analysis

Residual analysis studies the pattern, magnitude, and distribution of residuals. Randomly scattered residuals usually indicate a reasonable fit, whereas structured residuals may reveal trends, heteroskedasticity, omitted variables, or nonlinearity.

3.3.2 Outlier detection

Outlier detection uses discrepancy to identify observations that differ markedly from the rest of the data. An outlier may result from error, rare variation, or a meaningful but unusual case. Determining its cause often requires additional investigation rather than automatic removal.

4 Discrepancy in sampling and surveys

4.1 Sampling error and bias

In survey statistics, discrepancy can arise because a sample does not perfectly match the population. Random sampling error produces expected differences, while bias creates systematic departures caused by undercoverage, nonresponse, or faulty sampling procedures.

These two forms of discrepancy are treated differently. Sampling error is usually quantified and modeled, whereas bias requires design improvements or adjustment methods to reduce distortions.

4.2 Design-based discrepancy

Design-based approaches evaluate how closely a sample resembles the target population under the sampling design. Discrepancies may be examined in terms of demographic composition, response rates, or distribution of key variables. The goal is to determine whether the sample can support reliable estimates.

4.3 Weighting and adjustment

Survey weights and adjustment procedures are often used to reduce discrepancy between sample data and population benchmarks. Weighting can compensate for unequal selection probabilities or differential nonresponse, while calibration methods align sample totals with known population totals.

These adjustments do not eliminate all discrepancy, but they can improve representativeness and reduce certain forms of error.

5 Discrepancy in experimental design

5.1 Balance between groups

In experiments, especially randomized studies, balance refers to the similarity of groups on observed characteristics. Discrepancy between groups may indicate unequal distribution of baseline variables, which can complicate interpretation of treatment effects.

Good design aims to minimize such differences so that outcome comparisons are more likely to reflect the treatment rather than preexisting imbalances.

5.2 Randomization checks

Randomization checks examine whether the allocation process produced groups that are approximately comparable. Some differences are expected by chance, so the purpose is not to require perfect equality but to identify unusual or important departures from balance.

These checks are often descriptive, since formal significance tests of baseline differences can be misleading in small or large samples.

5.3 Covariate imbalance

Covariate imbalance refers to discrepancy in background variables between experimental groups. It may arise even under proper randomization, particularly in small studies. Researchers may account for such imbalance through stratification, regression adjustment, or matched analyses.

6 Bayesian and computational uses

6.1 Discrepancy functions

In Bayesian analysis, a discrepancy function is a chosen summary that measures how unusual the observed data are under a model. The function may be based on residuals, counts, extremes, or other features relevant to the scientific question.

Because the function is selected by the analyst, it reflects the kind of departure that is considered important in the application.

6.2 Posterior predictive checks

Posterior predictive checks compare observed data with data simulated from the posterior predictive distribution. The discrepancy between them helps assess whether the model can generate data resembling what was actually observed.

These checks are valued for their flexibility. They can be tailored to particular features of interest, such as tails, cluster sizes, or dependence structures.

6.3 Simulation-based assessment

Computational methods often evaluate discrepancy through repeated simulation. By generating data under a model and comparing simulated discrepancies with observed ones, analysts can judge whether the observed data look typical or unusual.

This approach is widely used when analytic solutions are difficult or when models are too complex for simple closed-form diagnostics.

7.1 Residual

A residual is the observed value minus the fitted or expected value. It is one of the most common numerical expressions of discrepancy in statistical modeling.

7.2 Error

Error is a broader term that may refer to deviation from truth, uncertainty in estimation, or mistake in measurement. It overlaps with discrepancy but is often used in more specific technical ways.

7.3 Deviation

Deviation denotes departure from a central value, expectation, or norm. In statistics, it is commonly used for differences from a mean or other reference point.

7.4 Divergence

Divergence usually refers to a measure of difference between distributions or probability models. It is often more formal than discrepancy and may be defined through information-theoretic or geometric criteria.

</INTERNAL_LINK_CANDIDATES> Residual (observed minus fitted or expected value) Goodness-of-fit test (procedure for assessing agreement between data and a model) Hypothesis testing (formal method for evaluating a statistical claim) Model diagnostic (tool for checking model adequacy) Outlier (an observation unusually far from the rest of the data) Sampling error (random difference between a sample and its population) Sampling bias (systematic distortion in a sample) Survey weighting (adjustment of survey data to improve representativeness) Calibration weighting (weighting method aligning sample totals with known benchmarks) Randomization (allocation process used to create comparable experimental groups) Covariate imbalance (unequal distribution of background variables across groups) Posterior predictive distribution (Bayesian distribution for simulated future or replicated data) Posterior predictive check (Bayesian model check using simulated data) Residual analysis (study of residual patterns to assess fit) Mahalanobis distance (multivariate distance accounting for covariance) Chi-square statistic (aggregate measure of deviation between observed and expected counts) Standard error (measure of sampling variability around an estimate) Probability distribution (model describing the likelihood of possible values) Model misspecification (failure of a model to match the data-generating process) Deviation (departure from a reference value)