What does Cronbach's alpha mean?

Cronbach's alpha measureshow well a set of items (or variables) measures a single unidimensionallatent construct.  When data have a multidimensional structure, Cronbach'salpha will usually be low. Technically speaking, Cronbach's alpha is nota statistical test - it is a coefficient of reliability (or consistency). Cronbach's alpha can be writtenas a function of the number of test items AND the average inter-correlationamong the items.  Below, for conceptual purposes, we show the formula for the standardized Cronbach's alpha:

Here N  is equal to the numberof items, c-bar is the average inter-item covariance among the items andv-bar equals the average variance.
One can see from this formula thatif you increase the number of items, you increase Cronbach's alpha. Additionally, if the average inter-item correlation is low, alpha willbe low.  As the average inter-item correlation increases, Cronbach'salpha increases as well.
This makes sense intuitively - ifthe inter-item correlations are high, then there is evidence that the itemsare measuring the same underlying construct. This is really what is meantwhen someone says they have "high" or "good" reliability.  They arereferring to how well their items measure a single unidimensional latentconstruct.
Thus, if you have multi-dimensionaldata, Cronbach's alpha will generally be low for all items.  In thiscase, run a factor analysis to see which items load highest on which dimensions,and then take the alpha of each subset of items separately.
An exampleLet's work through an example of howto compute Cronbach's alpha using SPSS, and how to check the dimensionalityof the data using factor analysis.
First, let's start with a dataset consistingof four test items - q1, q2, q3, and q4

        DATA LIST FREE
/q1 q2 q3 q4.
        BEGIN DATA.
2 3 5 5
5 5 4 4       
4 5 5 5
4 3 4 4
         3 3 5 5
3 3 4 5
3 4 4 4       
4 4 5 5
4 5 5 5
         4 4 3 3
4 4 5 5
5 5 4 4       
4 4 4 4
4 3 5 5
         4 4 5 5
3 3 4 5
4 5 4 4       
5 5 5 5
5 5 4 4
         4 4 4 4
4 4 4 4
4 4 4 4       
3 4 5 5
5 3 5 5
         5 5 3 3
3 3 4 4
4 4 4 4       
3 3 5 5
4 4 3 3
         4 4 5 5
4 4 5 5
4 5 5 5       
4 4 5 5
4 5 5 5
         4 4 5 5
3 3 4 4
4 3 5 4       
3 4 5 5
4 4 5 4
         3 4 4 4
4 5 5 5
5 5 5 5       
4 4 5 5
4 4 4 4
         4 4 5 5
3 4 4 4
5 5 5 5       
4 5 4 4
3 4 4 4
         5 3 4 4
5 3 4 4
4 5 4 4       
2 5 5 5
3 4 5 5
         4 3 5 5
4 4 4 4
4 4 5 5       
3 4 4 4
4 4 5 4
         4 4 5 5
END DATA.

To compute Cronbach's alphafor all four items - q1, q2, q3, q4 - use the reliabilitycommand:

        RELIABILITY
  /VARIABLES=q1 q2 q3 q4.

Here is the resulting outputfrom the above syntax:

R E L I A B I L I T Y   A N A L Y S I S  -   S C A L E   (A L L)        Reliability Coefficients       
N of Cases =     60.0                   N of Items =  4       
Alpha =    .3924

Here, the reliability isshown to be low using all four items because alpha is .3924. (Notethat a reliability coefficient of .70 or higher is considered "acceptable" in most Social Science research situations). Perhaps the data aremultidimensional? To check the dimensionality of the data, use the factorcommand:

        FACTOR
  /VARIABLES q1 q2 q3 q4        
  /FORMAT SORT BLANK(.35).

Here is the resulting outputfrom the above syntax:Total Variance Explained
Initial EigenvaluesExtraction Sumsof Squared LoadingsRotation Sums ofSquared LoadingsComponentTotal% of VarianceCumulative %Total% of VarianceCumulative %Total% of VarianceCumulative %11.92648.13948.1391.92648.13948.1391.89047.24747.24721.30632.64280.7811.30632.64280.7811.34133.53480.7813.65416.34997.130





4.1152.870100.000





ExtractionMethod: Principal Component Analysis.
Rotated Factor Loadings(a)
Factor12Q3.968
Q4.967
Q2
.827Q1
.807ExtractionMethod: Principal Component Analysis.
Rotation Method: Varimax with Kaiser Normalization. a Rotationconverged in 3 iterations.Notice that the data arenot unidimensional.  That is, q3 and q4 do not seeminglymeasure the same latent construct as q1 and q2.
Now, let's estimate the reliabilityof these two "subsets" of items separately:

        RELIABILITY
  /VARIABLES=q1 q2 q3 q4       
  /SCALE(Q1_Q2)=q1 q2
  /SCALE(Q3_Q4)=q3 q4.

Here is the resulting outputfrom the above syntax:

    R E L I A B I L I T Y   A N A L Y SI S   -   S C A L E   (Q 1 _ Q 2)        Reliability Coefficients       
N of Cases =     60.0                   N of Items =  2       
Alpha =    .5045       
  R E L I A B I L I T Y   A N A L Y S I S  -   S C A L E   (Q 3 _ Q 4)       
Reliability Coefficients       
N of Cases =     60.0                   N of Items =  2       
Alpha =    .9368

Here we see that the reliabilityfor items q3 and q4 is very high, while the reliability forq1and q2 is lower.  Both estimates of reliability, however, arehigher than when using all four items for measuring the same construct. This result also implies that the correlation between items q3 andq4is higher than the correlation between q1 and q2.  Tocheck that this is indeed true, use the correlations command:

        CORRELATIONS VARIABLES=q1 q2 q3 q4.

Correlations
Q1Q2Q3Q4Q1Pearson Correlation1.000.337-.084-.168Sig. (2-tailed)..008.525.200N60606060Q2Pearson Correlation.3371.000.001-.037Sig. (2-tailed).008..992.777N60606060Q3Pearson Correlation-.084.0011.000.881Sig. (2-tailed).525.992..000N60606060Q4Pearson Correlation-.168-.037.8811.000Sig. (2-tailed).200.777.000.N60606060
Here, we see that the corr(q1,q2)= .337, while the corr(q3,q4) = .881, confirming the resultsof the reliability analysis.  This higher correlation for q3and q4 can also be illustrated by the higher rotated factor loadingsof the exploratory factor analysis.Thus, since the data are not unidimensional,all four items SHOULD NOT be combined to create one single scale. Instead, q1 and q2 should be combined to create one scale,and q3 and q4 should be combined to create a second scale.