Cronbach's Alpha

Cronbach's Alpha is a measure of the internal consistency of a group of questions. The group of questions is called a scale, and each question in the group is an item. Under suitable assumptions, alpha can be used to estimate the reliability of the scale scores.

Cronbach's Alpha is calculated from the relationships among the item scores. It is not itself a simple correlation. Alpha is usually between 0 and 1, but it can be negative when items are negatively related on average, for example because a reverse-worded item was not recoded. The higher the average correlation between items, the greater the internal consistency of a test, provided that the items are intended to measure the same construct.

Latent variables

Hypotheses often contain variables that cannot be measured directly. Variables that are not directly measurable are called latent variables and are, for example, writing ability, intelligence or the attitude toward electric cars.

In order to make latent variables "measurable", a scale is used. A scale is a group of questions used to collectively measure a latent variable.

Items intended to measure the same construct should generally be related. However, they do not need to correlate as highly as possible: extremely similar items may simply be redundant.

Reliability and Cronbach's Alpha

If the answers to the questions or items are highly correlated, this is called high internal consistency. It is this internal consistency that Cronbach's Alpha measures.

Reliability describes the consistency of scores and the proportion of score variation that is not due to measurement error. The less measurement error there is, the more reliable the scores are.

Cronbach's Alpha therefore measures the extent to which a group of items is related. Under the assumptions described below, it provides an estimate of the reliability of the total or mean score formed from those items.

Assumptions for Cronbach's Alpha

Software can calculate Cronbach's Alpha without first checking its assumptions, but interpreting it as a reliability estimate requires assumptions. In classical test theory, the most important are:

• The measurement errors of different items should be uncorrelated.
• The items should measure one construct and be essentially tau-equivalent: their true scores may differ by a constant, but each item should measure the construct on the same scale. When this assumption is not reasonable, alternatives such as McDonald's omega may be more appropriate.

These assumptions may not hold exactly in practice. Furthermore, alpha tends to increase when more similarly related items are added, so a high alpha can partly reflect the number of items rather than exceptionally strong relationships among them.

It is important to note that Cronbach's Alpha does not test whether the items measure one or several latent variables. A high alpha is not evidence that a scale is unidimensional. The scale's dimensionality should be examined separately, for example with factor analysis.

For the reliability of the scale to be estimated using Cronbach's Alpha, the condition that all questions or items measure the same latent variable must be met!

In other words, if all items measure the same latent variable and the other assumptions are reasonable, Cronbach's Alpha estimates the reliability of the scale score. It does not show that the scale is valid or measures the intended construct.

Calculate Cronbach's Alpha

Cronbach's Alpha can be calculated using the following formula:

Cronbach's Alpha generally increases as the number of items or the average inter-item correlation increases. It becomes smaller when the average inter-item correlation decreases.

Example Cronbach's Alpha

Let's say your hypothesis is: Extroverts earn more than introverts. How do you measure salary? That is easy! Just ask in the questionnaire!

But how is extraversion measured in people? Through a literature research you have discovered that Extraversion can be measured by the following scale from the Big Five Personality Traits.

So, you create a survey on numiqo.com, send it out and get the answers in an Excel spreadsheet.

The sample dataset can be downloaded here.

The four variables can now be combined into a construct that gives you a value for your unmeasurable latent variable. For example, you could do this with a sum index or a mean index.

Before that, we examine whether the items are sufficiently consistent to be combined. Cronbach's Alpha is one part of this assessment; the wording and dimensionality of the items should also be considered.

This is done by copying the data into the upper table of the Cronbach's Alpha calculator. Then the four items are selected and numiqo calculates the reliability statistics.

For the present data a Cronbach's Alpha of 0.71 was obtained. The table of item scale statistics is then displayed. In the table you can see how the Cronbach's Alpha changes when the respective variable or item is omitted.

It can be seen that when item 1 is removed, Cronbach's alpha drops to 0.66, and when item 2 is removed, it drops to 0.48. However, when item 4 is removed, alpha increases to 0.79. This is a reason to inspect item 4, including whether it requires reverse coding, but not an automatic reason to remove it. The item's content and importance to the construct must also be considered.

Interpret Cronbach's Alpha

Rules of thumb such as 0.7 for an acceptable alpha are common, but they are not universal cutoffs. The required reliability depends on the purpose of the scale, the consequences of measurement error, and whether decisions concern groups or individuals. An alpha much higher than 0.9 can indicate redundant questions, although this should be checked from the item content and inter-item correlations. The table below is therefore only a rough guide.

As mentioned above, internal consistency describes relationships among the items, but it does not establish that they fit together in terms of content or that the scale is valid. The researcher must therefore use substantive knowledge and evidence about the scale's dimensionality and validity as well.

Cronbach's Alpha increases with the number of items. For example, if the scale is constructed with 8 items rather than 4, then the same correlation for the 8 items will tend to result in a larger alpha.

Positively and negatively worded questions can be used in the same scale, but reverse-worded items must be recoded before calculating alpha so that a high score has the same meaning for every item.

For ordinal response items with only a few categories, alpha based on ordinary Pearson correlations may be less suitable. An ordinal alpha based on polychoric correlations or an ordinal reliability coefficient can then be considered. Because alpha is estimated from a sample, reporting a confidence interval is also more informative than reporting only the point estimate, especially with a small sample.