The term "Cronbach's alpha" itself is the wrong name giving false information about the formula's original author. I will discuss history later and concentrate on mathematical properties here.

In most textbooks, alpha is described as a reliability coefficient. However, it is difficult to find the literature describing what conditions the data must meet to make sure that alpha equals reliability. Therefore, it is easy to misunderstand that alpha will always have the same value as reliability, regardless of whether the condition is met or not.

The necessary and sufficient condition for alpha to equal reliability is that the data are tau-equivalent (for simplicity, we assume the data are uni-dimensional and the errors are independent of each other). Alpha is less than reliability when the data is not tau-equivalent. The term tau-equivalence can be unfamiliar to readers. Please refer to the variance-covariance matrix below.

All parallel data are also tau-equivalent. Therefore, alpha can be used as a reliability coefficient if it is applied to parallel data and tau equivalent data. However, if applied to data that is not tau-equivalent, alpha has a lower value than reliability. The example presented in the congeneric data is the variance-covariance matrix of the data I presented to disprove that "if there is no measurement error, Cronbach's alpha has a value of one". In other words, when applying alpha to data that violates the prerequisite of alpha, alpha is less than 1, even though there is no measurement error.

Are you disappointed that alpha is not an unbiased estimator of reliability? In fact, producing consistently smaller estimates than reliability is a not-so-bad property as an estimator of reliability. You can be sure that the actual reliability is larger than the value of alpha. Therefore, there are many claims that it is more accurate to call alpha as a lower bound of reliability than to call it a reliability coefficient.

Do you understand what the lower bound of reliability means? Let's take an example. If the alpha value is .7, what is the actual reliability value? So far you have probably thought of a reliability value of .7. It is wrong. If the alpha value is .7, the reliability value can be .7 or greater than .7. However, it is not smaller than .7.

Then, what reliability coefficient should be used when the data are not tau-equivalent? I will explain this next.

For additional reference, please see:

Cho, E., & Kim, S. (2015). Cronbach's coefficient alpha: Well known but poorly understood.

*Organizational Research Methods*, 18, 207-230.