Surveys are constructed from carefully chosen questions to gather information from some group of people. Quite often, it’s not enough to just ask a single question. Surveys usually consist of multiple related questions that tries to measure a complex human behavior or characteristic, known as constructs. When the responses to these individual questions are taken together, they can be combined to form a score or scale. A survey is reliable if it gives similar results when taken again by a similar group of people.
Cronbach’s Alpha is a way to measure internal consistency reliability of the individual questions (or test items):
\[\alpha = \frac{N\bar{c}}{\bar{v} + (N-1)\bar{c}}\], where $\bar{c}$ is the average of the inter-item covariance, and $\bar{v}$ is the average variance.
Let’s consider this sample data, where X1, X2, X3, and X4 are questions that take on integer values from 1 to 5 (Likert). So, all these questions form a scale. There are 10 rows, so 10 people filled out this survey.
\[\begin{array}{|c|c|c|c|c|} \hline \text{id} & X1 & X2 & X3 & X4 \\ \hline 1 & 3 & 3 & 3 & 3 \\ 2 & 3 & 3 & 4 & 3 \\ 3 & 3 & 3 & 3 & 1 \\ 4 & 5 & 4 & 4 & 5 \\ 5 & 2 & 1 & 4 & 4 \\ 6 & 4 & 4 & 4 & 5 \\ 7 & 3 & 2 & 3 & 3 \\ 8 & 3 & 4 & 3 & 3 \\ 9 & 4 & 2 & 2 & 3 \\ 10 & 3 & 3 & 2 & 3 \\ \hline \end{array}\]Let’s find $\bar{c}$. The covariance matrix is:
\[\begin{array}{cccc} & \text{X1} & \text{X2} & \text{X3} & \text{X4} \\ \text{X1} & 0.6778 & 0.478 & 0.0444 & 0.456 \\ \text{X2} & 0.4778 & 0.989 & 0.1333 & 0.256 \\ \text{X3} & 0.0444 & 0.133 & 0.6222 & 0.489 \\ \text{X4} & 0.4556 & 0.256 & 0.4889 & 1.344 \\ \end{array}\]So, $\bar{c} = (0.478 + 0.0444 + 0.1333 + 0.456 + 0.256 + 0.489)/6 = 0.309$, and $\bar{v} = (0.6778 + 0.989 + 0.6222 + 1.344)/4 = 0.908$.
Then,
\[\alpha = \frac{4*0.309}{0.908 + 3*0.309} = 0.674\]A quick check in R gives the same result:
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Cronbach's alpha for the 'df' data-set
Items: 4
Sample units: 10
alpha: 0.674
How do you use this value? According to George and Mallery (2003), an $\alpha$ >= 0.9 is excellent, >= 0.8 is good, >= 0.7 is acceptable, >= 0.6 is questionable, >= 0.5 is poor, and < 0.5 is unacceptable.