# Cronbach's Alpha

Cronbach's Alpha (or tau-equivalent reliability) is a measure of the relationship between a group of questions. The group of questions is called a scale and each question in the group is an item. Cronbach's alpha is therefore a measure of the internal consistency of a scale and therefore of the strength of its reliability.

Cronbach's Alpha is the correlation between the answers in a questionnaire and can take values between 0 and 1. The higher the average correlation between items, the greater the internal consistency of a test.

## 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.

The goal now is that the answers to the different items match well, i.e. correlate highly. Each individual question should correlate as highly as possible with every other question.

## 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.

### Definition Cronbach's Alpha

Cronbach's Alpha is a measure of the internal consistency of a scale.

Reliability indicates how reliably or accurately a questionnaire or test measures a true value. Reliability therefore means how accurately a test can measure a variable. The less measurement error there is, the more reliable a test is.

Cronbach's Alpha is therefore a measure of the extent to which the group of questions are related to each other and thus provides an estimate of how good or poor the measurement accuracy, known as reliability, of a group of items is.

## Assumptions for Cronbach's Alpha

In the context of classical test theory, the focus is on the measurement errors that exist when a value is measured. In order to calculate Cronbach's Alpha, two conditions must be met.

- The error proportions of the items must be uncorrelated, i.e. the error proportion of one item must not be influenced by the error proportion of another item.
- The items must have the same proportion of true variance.

In practice, however, neither of these conditions is usually met. Furthermore, the more items a scale has, the higher the alpha value will be.

It is important to note that Cronbach's Alpha does not test whether each item is actually influenced by only one or more latent variables! A high value of is not evidence that the items are influenced by only one latent variable.

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 the items measure the same latent variable, then Cronbach's Alpha tells us how well these items measure the latent variable.

## Calculate Cronbach's Alpha

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

Cronbach's Alpha therefore increases as the number of items increases and as the inter-item correlation increases. correlation between the items increases. Cronbach's Alpha becomes smaller when the average inter-item correlation becomes smaller.

## 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 datatab.de, 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, of course, we need to check to what extent these items represent the same thing, i.e., how high Cronbach's Alpha is and how reliable the scale is.

This is done by copying the data into the upper table of the Cronbach's Alpha calculator. Then the four items are selected and DATAtab 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, the Cronbach's alpha drops to 0.66 and when item 2 is removed, the Cronbach's alpha even drops to 0.48. However, when item 4 is removed, the Cronbach's alpha increases to 0.79. Therefore, in this case it could be considered to remove item 4.

## Interpret Cronbach's Alpha

Cronbach's Alpha should not be less than 0.6. Values above 0.7 are considered acceptable. However, the Cronbach's Alpha should preferably not be much higher than 0.9, as this would mean that the questions are "too similar" and therefore you get the same answers to the questions, in which case you could omit questions that are too highly correlated and you would not have any loss of information. The table below can be used to interpret Cronbach's Alpha.

Cronbach's Alpha | Interpretation |
---|---|

> 0,9 | excellent |

> 0,8 | good |

> 0,7 | acceptable |

> 0,6 | questionable |

> 0,5 | poor |

< 0,5 | unacceptable |

As mentioned above, internal consistency only says something about the correlation of the items, but not about whether the items fit together in terms of content. Cronbach's Alpha only checks whether the items are correlated. The researcher must therefore ensure that only items that measure the same content are used.

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.

Furthermore, it is also important to ensure that the questions are all formulated in either a positive or a negative way. That is, a high or low value must always mean the same thing.

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