{ "id": "epsd2yre47", "slug": "mean-centering", "title": "Mean centering", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" } ], "geo": [ { "id": "geo.nongeo.t1800.physical.math.statistics", "label": "Non-geographic · 1800 to 1945: Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical" }, { "id": "geo.nongeo.t1800.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math" }, { "id": "geo.nongeo.t1800.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics" } ] } ], "excerpt": "Mean centering is a statistical preprocessing step that subtracts a variable's mean from each observation, giving transformed values that average to zero; it is used in regression and PCA.", "snippet": "Mean centering is a statistical preprocessing step that subtracts a variable's mean from each observation, giving transformed values that average to zero; it is used in regression and PCA.", "node": "physical.math.statistics", "markdown": "# Mean centering\n\nMean centering is a data preprocessing step that subtracts a variable's mean from every observation on that variable, so the transformed values average to zero; IUPAC defines the operation as \\( a_{i,j}^{*} = a_{i,j} - \\tfrac{1}{n}\\sum_{i=1}^{n} a_{i,j} \\).[1](https://goldbook.iupac.org/terms/view/10053) More generally, centering means selecting a reference value for each predictor and coding the data so that each estimated coefficient is relevant to the research question.[2](https://doi.org/10.1002/mpr.170) It is used in moderated regression, multilevel modeling, and multivariate methods such as principal component analysis, partial least squares, and discriminant analysis.[1](https://goldbook.iupac.org/terms/view/10053) • [3](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2025.1634152/full)\n\n| Key fact | Detail |\n|---|---|\n| Operation | Subtract the variable mean from each observation; transformed data have mean zero [1](https://goldbook.iupac.org/terms/view/10053) |\n| Correlation with product terms | Under bivariate normality, centered predictors have zero correlation with their centered product [4](https://jse.amstat.org/v19n3/afshartous.pdf) |\n| What does not change | \\( R^2 \\), adjusted \\( R^2 \\), and the highest-order interaction coefficient and standard error are identical with or without centering [5](https://www.bauer.uh.edu/jhess/papers/JMRMeanCenterPaper.pdf) |\n| Multilevel choice | Grand-mean and group-mean centering generally produce estimates that differ in both value and meaning [6](https://doi.org/10.1037/1082-989x.12.2.121) |\n| Versus standardizing | Centering changes what zero means; standardizing additionally divides by the standard deviation [7](https://uoepsy.github.io/lmm/10_centering.html) |\n| Multivariate caveat | Recommended for PCA, PLS, and discriminant analysis, but incompatible with non-negativity constraints such as in multivariate curve resolution [1](https://goldbook.iupac.org/terms/view/10053) |\n\n## How it works\n\nIn a moderated regression \\( y = b_0 + b_1 x_1 + b_2 x_2 + b_3 x_1 \\cdot x_2 + e \\), the first-order coefficients are conditional (simple) effects evaluated at zero on the other predictor, not main effects, whether or not the data are centered.[3](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2025.1634152/full) Centering redefines where zero sits: if \\( x_2 \\) is centered at its mean, \\( b_1 \\) becomes the effect of \\( x_1 \\) when \\( x_2 = \\mu_2 \\) instead of when \\( x_2 = 0 \\).[4](https://jse.amstat.org/v19n3/afshartous.pdf) This is why lower-order coefficients and the intercept change after centering while the model itself does not.\n\nThe covariance algebra explains the collinearity effect: \\( \\mathrm{Cov}(x_1 \\cdot x_2,\\, x_1) = \\sigma_1^2 \\mu_2 + \\mathrm{Cov}(x_1, x_2)\\,\\mu_1 \\), so when both predictors have mean zero, a main effect has zero correlation with its product term only under additional conditions such as joint normality, since the covariance is exclusively a function of a central third-order moment of the joint distribution.[4](https://jse.amstat.org/v19n3/afshartous.pdf) • [19](https://pmc.ncbi.nlm.nih.gov/articles/PMC6713984/) Under bivariate normality, each centered predictor has zero covariance, and therefore zero correlation, with the centered product \\( (x_1 - \\mu_1) \\cdot (x_2 - \\mu_2) \\); any remaining correlation is due solely to nonnormality.[4](https://jse.amstat.org/v19n3/afshartous.pdf)\n\nA worked example shows what stays fixed. Uncentered (\\( \\hat{y} = 657.69 - 8.58x - 55.65\\,\\mathrm{mod} + 0.96\\,xm \\)) and centered (\\( \\hat{y} = 203.02 + 1.02\\,x_c + 39.88\\,\\mathrm{mod}_c + 0.96\\,x_c \\cdot \\mathrm{mod}_c \\)) regressions give the same interaction coefficient (0.96), standard error (0.265), t-value (3.626), p-value (.000), confidence interval [0.434, 1.486], and model summary.[8](https://study.sagepub.com/sites/default/files/gurnsey_onlineapp17.pdf) The centered slope 1.02 can be recovered from the uncentered equation as the slope at the moderator's mean, \\( b = b_x + b_{xm} \\cdot \\mathrm{mod} = -8.58 + 0.96 \\cdot 10 = 1.02 \\).[8](https://study.sagepub.com/sites/default/files/gurnsey_onlineapp17.pdf)\n\n## How it is done\n\nPractically, mean centering is a two-step procedure: calibration estimates the column means from a calibration data set \\( X \\), and application subtracts those means from the data.[9](https://eigenvector.com/wp-content/uploads/2020/01/Data-preprocessing-objective.pdf) In regression the main choice is the reference value. Grand-mean centering subtracts the full-sample mean; group-mean centering subtracts each observation's cluster mean.[10](https://web.pdx.edu/~newsomj/mlrclass/ho_centering.pdf)\n\nOne published default strategy codes binary variables as +1/2, ordinal variables as deviations from their median, and dummy variables for \\( m \\) categories as \\( 1 - 1/m \\) and \\( -1/m \\), then computes interaction terms from the centered predictors.[2](https://doi.org/10.1002/mpr.170) Centering differs from standardization (z-scoring), which subtracts the mean and then divides by the standard deviation, changing the unit a slope represents while leaving its significance unchanged; re-centering a linear-model predictor changes only the intercept, that is, what zero means.[7](https://uoepsy.github.io/lmm/10_centering.html)\n\n## Origin\n\nThe modern literature on centering in interactive models runs through a series of methodological papers. Richard L. Tate (1984, *Sociological Methods & Research*) examined the limitations of centering for interactive models and framed the centered product \\( (x_1 - d) \\cdot (x_2 - e) \\), being uncorrelated with both \\( x_1 \\) and \\( x_2 \\), as a residualized \\( x_1 \\cdot x_2 \\) term.[11](https://doi.org/10.1177/0049124184013002004) The recommendation to center predictors entering higher-order interactions, with a computational rationale and the zero-covariance result under bivariate normality, traces to the 1991 book *Multiple Regression: Testing and Interpreting Interactions* by Leona S. Aiken, Stephen G. West, and Raymond R. Reno.[5](https://www.bauer.uh.edu/jhess/papers/JMRMeanCenterPaper.pdf) Kreft, de Leeuw, and Aiken (1995) analyzed the forms of centering in hierarchical linear models in *Multivariate Behavioral Research* [12](https://doi.org/10.1207/s15327906mbr3001_1); Kraemer and Blasey (2004) proposed the default coding strategy described above in the *International Journal of Methods in Psychiatric Research* [2](https://doi.org/10.1002/mpr.170); Enders and Tofighi (2007) formalized the grand-mean versus within-cluster distinction for cross-sectional multilevel models in *Psychological Methods* [6](https://doi.org/10.1037/1082-989x.12.2.121); and Iacobucci and colleagues (2015) argued centering helps \"micro\" but not \"macro\" multicollinearity in *Behavior Research Methods*.[13](https://doi.org/10.3758/s13428-015-0624-x)\n\n## Variants\n\n**Multilevel centering** has two main forms with different meanings. Centering within cluster (CWC, group-mean centering) removes all between-cluster variation and yields a pure estimate of the pooled within-cluster Level 1 coefficient.[6](https://doi.org/10.1037/1082-989x.12.2.121) With group-mean centering plus the cluster mean reintroduced as a Level 2 predictor, one coefficient estimates the within-group effect and the other the between-group effect; with grand-mean centering plus a reintroduced mean, the second coefficient estimates the compositional effect, the difference between the between- and within-group slopes.[10](https://web.pdx.edu/~newsomj/mlrclass/ho_centering.pdf) Level 2 variables can only be grand-mean centered.[10](https://web.pdx.edu/~newsomj/mlrclass/ho_centering.pdf)\n\n**Matrix-level variants** include centering each data object at the origin so its entries have mean 0, grand mean centering, which subtracts the mean of all matrix entries, and double centering.[14](https://par.nsf.gov/servlets/purl/10439036)\n\n## Applications\n\nIn moderated regression, centering establishes a meaningful zero point so that first-order coefficients describe effects at typical predictor values rather than at zero, which often lies outside the data.[8](https://study.sagepub.com/sites/default/files/gurnsey_onlineapp17.pdf) • [2](https://doi.org/10.1002/mpr.170) In multilevel models it separates within- from between-cluster effects and, per published guidance for cross-level interactions, level-1 predictors are group-mean centered while level-2 predictors are grand-mean centered.[10](https://web.pdx.edu/~newsomj/mlrclass/ho_centering.pdf)\n\nIn PCA, partial least squares, and discriminant analysis, mean centering is generally recommended because relative values across samples matter more than absolute deviation from zero.[1](https://goldbook.iupac.org/terms/view/10053) After centering, the first principal component captures the most sum-of-squares about the mean, that is, variance about the mean, so the model must be interpreted differently in exploratory analysis.[9](https://eigenvector.com/wp-content/uploads/2020/01/Data-preprocessing-objective.pdf)\n\n## Limitations and alternatives\n\n**Centering does not fix essential collinearity.** [Correlation](https://www.edgepedia.org/correlation) due to scaling is nonessential multicollinearity and is removable by rescaling; correlation due to skew is essential and cannot be removed by rescaling.[4](https://jse.amstat.org/v19n3/afshartous.pdf) Because subtracting a constant leaves a variable's dispersion and its relationships to other predictors intact, centering can only affect nonessential collinearity.[3](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2025.1634152/full)\n\n**The correlation with the product term can increase.** For many jointly distributed random variables, even some with considerable symmetry, the correlation between centered main effects and their interaction can rise relative to the uncentered case; symmetry of lower-dimensional marginals is necessary but not sufficient for a decrease.[15](https://journals.sagepub.com/doi/10.1177/0013164418817801)\n\n**Interpretation risk.** Centering adds no new information and cannot produce more precise estimates; it only changes which question the coefficients and t-tests answer.[16](https://press.umich.edu/pdf/9780472099696-ch4.pdf)\n\n**Multilevel-specific findings.** Unlike OLS, where collinearity does not bias point estimates, uncentered level-1 predictors in multilevel models can give greatly biased slopes; disaggregation (cluster-mean centering plus cluster means as level-2 predictors) eliminates the possibility of fixed-effect bias from collinearity alone, though standard errors and random-effect covariances can still be biased under some conditions.[17](https://quantpsy.org/pubs/yaremych_preacher_2024.pdf)\n\n**Alternatives.** Residual centering, which orthogonalizes the product term against its components, biases the \\( x_1 \\) and \\( x_2 \\) effects and has been called \"a distinctly bad idea\" by critics who recommend addressing collinearity through design or larger samples instead [5](https://www.bauer.uh.edu/jhess/papers/JMRMeanCenterPaper.pdf); it remains useful for latent-variable interactions but carries caveats including perils of double orthogonalization, unintended effects on model fit, removal of a mean structure, and sensitivity to nonnormal data.[18](https://sage.cnpereading.com/doi/10.1177/0013164412445473) The 2025 review recommends a hierarchical (Step 1/Step 2) analysis as a default for assessing main effects and interactions without relying on centering.[3](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2025.1634152/full)\n\n## References\n\n1. [IUPAC Gold Book – mean centering](https://goldbook.iupac.org/terms/view/10053)\n2. [Helena C. Kraemer, Christine M. Blasey (2004). Centring in regression analyses: a strategy to prevent errors in statistical inference. International Journal of Methods in Psychiatric Research.](https://doi.org/10.1002/mpr.170)\n3. ['Mean centering is not necessary in regression analyses, and probably increases the risk of incorrectly interpreting coefficients' (Frontiers in Psychology, 2025)](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2025.1634152/full)\n4. [Afshartous & Preston, 'Key Aspects of Centering Predictor Variables' (Journal of Statistics Education, Vol. 19, No. 3, 2011)](https://jse.amstat.org/v19n3/afshartous.pdf)\n5. [Echambadi & Hess (2007), 'Mean-Centering Does Nothing for Moderated Multiple Regression' (Journal of Marketing Research; author-hosted PDF)](https://www.bauer.uh.edu/jhess/papers/JMRMeanCenterPaper.pdf)\n6. [Craig K. Enders, Davood Tofighi (2007). Centering predictor variables in cross-sectional multilevel models: A new look at an old issue.. Psychological Methods.](https://doi.org/10.1037/1082-989x.12.2.121)\n7. [10: Centering – LMM/MLM (University of Edinburgh course materials)](https://uoepsy.github.io/lmm/10_centering.html)\n8. [Gurnsey, online Appendix 17.2: Mean-Centering and Collinearity (SAGE textbook companion site)](https://study.sagepub.com/sites/default/files/gurnsey_onlineapp17.pdf)\n9. [Data preprocessing objective (Eigenvector Research)](https://eigenvector.com/wp-content/uploads/2020/01/Data-preprocessing-objective.pdf)\n10. [Centering in Multilevel Regression (Newsom, Portland State University class notes)](https://web.pdx.edu/~newsomj/mlrclass/ho_centering.pdf)\n11. [RICHARD L. TATE (1984). Limitations of Centering for Interactive Models. Sociological Methods & Research.](https://doi.org/10.1177/0049124184013002004)\n12. [Ita G.G. Kreft, Jan de Leeuw, Leona S. Aiken (1995). The Effect of Different Forms of Centering in Hierarchical Linear Models. Multivariate Behavioral Research.](https://doi.org/10.1207/s15327906mbr3001_1)\n13. [Dawn Iacobucci and colleagues (2015). Mean centering helps alleviate “micro” but not “macro” multicollinearity. Behavior Research Methods.](https://doi.org/10.3758/s13428-015-0624-x)\n14. [New Perspectives on Centering (NSF PAR)](https://par.nsf.gov/servlets/purl/10439036)\n15. [McClelland et al., 'Centering in Multiple Regression Does Not Always Reduce Multicollinearity' (Educational and Psychological Measurement, 2018/2019)](https://journals.sagepub.com/doi/10.1177/0013164418817801)\n16. [Kam & Franzese, 'Colinearity and Mean-Centering the Components of Interaction Terms' (University of Michigan Press chapter)](https://press.umich.edu/pdf/9780472099696-ch4.pdf)\n17. [Yaremych & Preacher (2024), 'Understanding the Consequences of Collinearity for Multilevel Models: The Importance of Disaggregation Across Levels' (Educational and Psychological Measurement; author PDF)](https://quantpsy.org/pubs/yaremych_preacher_2024.pdf)\n18. [Orthogonalizing Through Residual Centering (Educational and Psychological Measurement, 2012)](https://sage.cnpereading.com/doi/10.1177/0013164412445473)\n19. [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC6713984/)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://web.pdx.edu/~newsomj/mlrclass/ho_centering.pdf" ], "url": "https://www.edgepedia.org/mean-centering", "markdown_url": "https://www.edgepedia.org/mean-centering.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Mean centering\", Edgepedia (EdgeChat), https://www.edgepedia.org/mean-centering. Edgepedia Community License 1.0.", "credit_md": "\"[Mean centering](https://www.edgepedia.org/mean-centering)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/mean-centering](https://www.edgepedia.org/mean-centering). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Mean centering\", Edgepedia (EdgeChat), https://www.edgepedia.org/mean-centering. Edgepedia Community License 1.0.", "speakable": "Mean centering is a statistical preprocessing step that subtracts a variable's mean from each observation, giving transformed values that average to zero; it is used in regression and PCA." }