Editing Cronbach's alpha (section)

===The value of Cronbach's alpha ranges between zero and one===
By definition, reliability cannot be less than zero and cannot be greater than one. Many textbooks mistakenly equate <math>\rho_{T}</math> with reliability and give an inaccurate [[explanation]] of its range. <math>\rho_{T}</math> can be less than reliability when applied to data that are not essentially tau-equivalent. Suppose that <math>X_2</math> copied the value of <math>X_1</math> as it is, and <math>X_3</math> copied by multiplying the value of <math>X_1</math> by -1.

The covariance matrix between items is as follows, <math>\rho_{T}=-3</math>.
{| class="wikitable" style="text-align: right;"
|+ Observed covariance matrix
|-
!
! <math>X_1</math>
! <math>X_2</math>
! <math>X_3</math>
|-
! <math>X_1</math>
| <math>1</math>||<math>1</math>||<math>-1</math>
|-
! <math>X_2</math>
| <math>1</math>|| <math>1</math>|| <math>-1</math>
|-
! <math>X_3</math>
| <math>-1</math>|| <math>-1</math>|| <math>1</math>
|}
Negative <math>\rho_{T}</math> can occur for reasons such as negative discrimination or mistakes in processing reversely scored items.

Unlike <math>\rho_{T}</math>, SEM-based reliability coefficients (e.g., <math>\rho_{C}</math>) are always greater than or equal to zero.

This anomaly was first pointed out by Cronbach (1943)<ref name="c1943">{{cite journal|first=L. J.|last=Cronbach|title=On estimates of test reliability|journal=Journal of Educational Psychology|volume=34|issue=8|pages=485–494|date=1943|doi=10.1037/h0058608}}</ref> to criticize <math>\rho_{T}</math>, but Cronbach (1951)<ref name="Cronbach"/> did not comment on this problem in his article that otherwise discussed potentially problematic issues related <math>\rho_{T}</math>.<ref name="c2004"/>{{rp|396}}<ref>{{Cite journal |last1=Waller |first1=Niels |last2=Revelle |first2=William |date=2023-05-25 |title=What are the mathematical bounds for coefficient α? |url=https://doi.apa.org/doi/10.1037/met0000583 |journal=Psychological Methods |language=en |doi=10.1037/met0000583 |pmid=37227892 |issn=1939-1463|url-access=subscription }}</ref>