Cronbach's alpha equals reliability

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.

If there is no measurement error, Cronbach's alpha has a value of one

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.

If there is no measurement error, the reliability is one. Alpha is known as a reliability coefficient. Therefore, if there is no measurement error, it is easy to think that alpha also has a value of one.


Let us imagine a hypothetical case to make sure this proposition is true. I measured the height of one person three times, but I got the same value all three times. That is, the measurement error is zero. However, the first two measurements were recorded in cm (i.e., one-hundredth of a meter), and the last measurement was recorded in mm (i.e., one-thousandth of a meter). In the same way, I measured the height of ten people. The following data. 

Below is the SPSS analysis process and results. Unlike the expectation that a value of 1 would come out, we got a value of .438.

The above example shows that alpha is not a reliability coefficient as we think if its prerequisites are violated. If the prerequisite of being 'tau-equivalent' is violated, alpha is less than the reliability. I will later explain the meaning of being tau-equivalent.

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.

Cronbach's alpha has a value between zero and one

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.

Alpha is commonly known to have values ​​between zero and one. Some textbooks explain that the value of alpha ranges between 0 and 1, and some do not explain the range of alpha at all. Literature that explains that alpha can be negative is rare.

The notion that 'alpha has a value between zero and one' does not mean 'it is rare that alpha has a value less than zero', but 'it is impossible for alpha to have a value less than zero'.

I challenge the impossible. Let's do a simple analysis in SPSS for proof. The first variable is filled with random numbers. The second variable copies the first variable. The third variable multiplies the first variable by -1. The following is an example.

Below is the SPSS analysis process and results. Unlike what is commonly known, we got a value of -3.

Although the above example is artificial, the actual data analysis sometimes reports that alpha is negative. For example, a negative alpha value may be calculated if the score of the item measured by the inverse item is not reversed (e.g., 7-> 1, 6-> 2, ..., 1-> 7).

If you misunderstood that alpha always has a value greater than zero, you might have thought of the following:
(a) Reliability has a value between 0 and 1.
(b) Alpha equals reliability.
(c) Therefore, alpha has a value between 0 and 1.

The wrong premise here is 'alpha equals reliability'. Alpha is an estimator, or estimate, of reliability. If alpha's prerequisites are violated, alpha is not equal to reliability. Alpha may have a negative value if the prerequisites violate severely. The prerequisite for alpha, that is being "tau-equivalent", will be described later.

Not all reliability coefficients have this problem. Structural equation model-based reliability coefficients (e.g., composite reliability) always have a value between 0 and 1.

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.

How to obtain and use RelCalc

How to obtain RelcCalc

For any of the two links, do the followings.
1) Copy and paste the following link into the address bar of your web browser, 
2) press Enter key, 
3) and click the download button.

Program: 
https://drive.google.com/open?id=1mBNFtt3EeMr3cf-CncsO4rL3P_90LK9l
Tutorial: 

https://drive.google.com/file/d/1Hu_YUWp3j6Owig8LLZgS0G1oMLPnhc2y
AMOS data for tutorial: 
https://drive.google.com/open?id=0B04puHatlLV0RXJ1d2xabVplc0k
TXT data for tutorial: 
https://drive.google.com/open?id=0B04puHatlLV0VVZMZFo3N3g3bVk

If there is a problem, please send an email to me and I will send it to you directly. 

How to use RelCalc

The Microsoft Excel file consists of six worksheets. The first sheet briefly explains the measurement models. The second sheet examines the assumption of being tau-equivalent and parallel and calculates three unidimensional reliability coefficients. The third through sixth sheets calculate multidimensional reliability coefficients.

The second sheet requires the Solver Add-in to be activated. To activate it, 
1) click Excel Options, 
2) click Add-ins, 
3) find the Manage box in the drop list below and select Excel Add-ins, 
4) and click Go. 
5) In the Add-Ins available box, select the Solver Add-in and click OK. 
Once activated, you do not have to perform this process again.