# 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 that group is an item. Cronbach's alpha is therefore a measure of the internal consistency of a scale and thus of the strength of its reliability.

Cronbach's alpha is the correlation between the answers in a questionnaire. Like the correlation coefficient, Cronbach's alpha can take on values between -1 and 1. The internal consistency of a test is greater the greater the correlation between items on average.

## 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 for latent variables to be made "measurable," a scale is used. A scale is a group of questions that are 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. And it is precisely this internal consistency that Cronbach's alpha measures.

### Definition Cronbachs Alpha

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

Reliability indicates how reliably or precisely a questionnaire or test measures a true value. Reliability therefore means how accurately a test can measure a variable. The reliability of a test is higher the fewer measurement errors there are.

Cronbach's alpha is thus a measure of the extent to which the group of questions are related to one another 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 for Cronbach's alpha to be calculated, 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.

However, both conditions are usually not met in practice. Furthermore, the more items the scale has, the larger the alpha value becomes.

It is important to note that Cronbach's alpha does not test whether the individual items are really influenced by only one or by several latent variables! A high value of the Cronbach's alpha is no 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 fulfilled!

That is, if all items measure the same latent variable then the Cronbach's alpha tells how well these items measure the latent variable.

## Calculate Cronbach's alpha

The Cronbach's alpha can be calculated with this formula:

The Cronbach's alpha thus becomes larger when the number of items is increased and when the inter-item correlation increases. The 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 salary than introverts. So how is the salary measured? That is simple! You simply ask for it in the questionnaire!

But how is extraversion measured in people? Through research you have found 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 non-measurable latent variable. You can do this for example with a sum index or a mean index.

Before that, of course, we have 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.

For this purpose, the data are copied into the upper table in 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 with the 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 Itme 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

The Cronbach's alpha should not be below 0.6, if possible. Values above 0.7 are considered acceptable. However, the Cronbach's alpha should also preferably not be much greater than 0.9, which would mean that the questions are "too similar" and you therefore get the same answers to the questions, in which case you could omit questions that correlate too highly and you would not have any loss of information. For the interpretation of Cronbach's alpha, the table below can be used.

Cronbachs Alpha | Interpretation |
---|---|

> 0,9 | Excellent |

> 0,8 | Good |

> 0,7 | Acceptable |

> 0,6 | Questionable |

> 0,5 | Poor |

< 0,5 | Unacceptable |

As already written above, the internal consistency only says something about the correlation of the items, but not whether the items fit together in terms of content. Cronbach's alpha only checks whether the items correlate. Therefore, the researcher must ensure that only items that measure the same content are used.

The alpha increases with the number of items. For example, if the scale is not constructed with 4 items, but with 8 items, then the same correlation for the 8 items will tend to result in a larger alpha.

Furthermore, it must of course be ensured that the questions are all formulated either positively or negatively, i.e. a high or low value must always mean the same thing.

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