standardization instrument
| Definition | Subtracting a column's mean and dividing by its spread, so that it has mean zero and variance one. It is invertible and preserves every ordering, so it changes the reader and not the data. Chapter 2. Also rescal. |
|---|---|
| Example | Values 2, 4, and 6 with mean 4 and spread 1.633 become −1.22, 0, and 1.22. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id standardization, kind instrument. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 2.1. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/Standardization.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 2.1.
\[P_{C_2\circ C_1}(x)=J_1(x)^{\top}\,P_{C_2}\big(C_1(x)\big)\,J_1(x),\qquad \operatorname{rank}P_{C_2\circ C_1}(x)\le\operatorname{rank}P_{C_2}\big(C_1(x)\big)\quad\text{at each row } x.\]
Book equation 0.6.
\[x_{\mathrm w}=\Sigma^{-1/2}(x-\mu),\qquad \Sigma^{-1/2}=\sum_i\lambda_i^{-1/2}\,v_i v_i^{\top}.\]
Conditions
- Subtracting a column’s mean and dividing by its spread. The standardized column has weighted mean zero and weighted variance one, the transform is invertible, and it preserves every ordering of the rows, so it changes the reader’s geometry and not what the table holds.
- An invertible transform identifies no two rows and forms no quotient. For a consumer that reads distances across columns the declaration that scale is irrelevant is almost always right, and for a consumer that reads one column it does nothing.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 2 section 2.5 of Data Mining as Observation, with
the whitened code’s record in
geometric-observation/chapters/ch08_value.md:84-97.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/Standardization.lean,
theorems mean_zero, variance_one,
standardize_inv, standardize_lt_iff, at
observation-data-mining f3914f0; what the check covers is stated in the
book’s appendix
C.
Used in
Data Mining as Observation primers L and S, chapters 0, 1, 2, 3, 4, 6, 9, 10, 11.
Related
whitening; imputation; Euclidean distance; quotient; outlier.
See also
Book equations stated beside the entry’s terms, not defining it: 3.1.
Sources-table rows that share a record with the entry without naming it: chapter 2 section 2.1, chapter 2 section 2.5, chapter 4 section 4.3.
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.