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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.

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