Why is my Cronbach's $\alpha$ negative

Question

I'm just learning about Cronbach's $\alpha$. I have recently tried to carry out this test. Having 80 completed questionnaires with 15 questions (variables), the questionnaire used the Likert scale 1-5 (1 strongly agree) I have filled the data in, however I've ended up with a negative result! What might I have done wrong?

Answer 1

Recall how cronbachs-alpha is calculated: assuming that a total test score $X$ is a sum of scores on $K$ separate items, $X=Y_1+\dots+Y_K$, then

$$\alpha:=\frac{K}{K-1}\left(1-\frac{1}{\sigma_X^2}\sum_{i=1}^K\sigma_{Y_i}^2\right),$$

where $\sigma_{Y_i}^2$ and $\sigma_X^2$ denote the variance of the $i$-th item and the total score, respectively.

In theory, $0\leq\alpha\leq 1$. In practice, you will substitute estimates of $\sigma_{Y_i}^2$ and $\sigma_X^2$, and you can definitely end up with negative values.

Now, this means that your items quite definitely do not measure the same underlying construct. Nick Cox already gave some very good points: you may need to reverse the coding of some items, or something similar. It would be good if you went and thought very critically about the wording of your items and try to make connections between them. It's probably indeed best to start by looking at correlations between items.