Cronbach’s alpha reliability coefficient normally ranges between 0 and 1. However, there is actually no lower limit to the coefficient. The closer Cronbach’s alpha coefficient is to 1.0 the greater the internal consistency of the items in the scale.

Based upon the formula _ = rk /[1 + (k -1)r] where k is the number of items considered and r is the mean of the inter-item correlations the size of alpha is determined by both the number of items in the scale and the mean inter-item correlations. George and Mallery (2003) provide the following rules of thumb:

“_ > .9 – Excellent,

_ > .8 – Good,

_ > .7 – Acceptable,

_ > .6 – Questionable,

_ > .5 – Poor, and

_ < .5 – Unacceptable” (p. 231).

While increasing the value of alpha is partially dependent upon the number of items in the scale, it should be noted that this has diminishing returns. It should also be noted that an alpha of .8 is probably a reasonable goal. It should also be noted that while a high value for Cronbach’s alpha indicates good internal consistency of the items in the scale, it does not mean that the scale is unidimensional.

When using Likert-type scales it is imperative to calculate and report Cronbach’s alpha coefficient for internal consistency reliability for any scales or subscales one may be using. The analysis of the data then must use these summated scales or subscales and not individual items. If one does otherwise, the reliability of the items is at best probably low and at worst unknown.

Cronbach’s alpha does not provide reliability estimates for single items.

where K is the number of components (K-items or testlets),  the variance of the observed total test scores, and  the variance of component i for the current sample of persons. See Develles (1991).

Alternatively, the Cronbach’s α can also be defined as

where K is as above,  the average variance, and  the average of all covariances between the components across the current sample of persons.

The standardized Cronbach’s alpha can be defined as

where K is as above and  the mean of the K(K − 1) / 2 non-redundant correlation coefficients (i.e., the mean of an upper triangular, or lower triangular, correlation matrix).

Cronbach’s α is related conceptually to the Spearman–Brown prediction formula. Both arise from the basic classical test theory result that the reliability of test scores can be expressed as the ratio of the true-score and total-score (error plus true score) variances:

Theoretically, alpha varies from zero to 1, since it is the ratio of two variances. Empirically, however, alpha can take on any value less than or equal to 1, including negative values, although only positive values make sense. Higher values of alpha are more desirable. Some professionals^[3] as a rule of thumb, require a reliability of 0.70 or higher (obtained on a substantial sample) before they will use an instrument. Obviously, this rule should be applied with caution when α has been computed from items that systematically violate its assumptions.^[specify] Furthermore, the appropriate degree of reliability depends upon the use of the instrument. For example, an instrument designed to be used as part of a battery of tests may be intentionally designed to be as short as possible, and therefore somewhat less reliable. Other situations may require extremely precise measures with very high reliabilities. This has resulted in a wide variance of test reliability. In the case of psychometric tests, most fall within the range of 0.75 to 0.83 with at least one claiming a Cronbach alpha above 0.90