One problem with the split-half method is that the reliability estimate obtained using any random split of the items is likely to differ from that obtained using another. One solution to this problem is to compute the Spearman-Brown corrected split-half reliability coefficient for every one of the possible split-halves and then find the mean of those coefficients. This is the motivation for Cronbach’s alpha.
Cronbach’s alpha is superior to Kuder and Richardson Formula 20 since it can be used with continuous and non-dichotomous data. In particular, it can be used for testing with partial credit and for questionnaires using a Likert scale.
Definition 1: Given variable x1, …, xk and x0 = and Cronbach’s alpha is defined to be
Property 1: Let xj = tj + ej where each ej is independent of tj and all the ej are independent of each other. Also let x0 = and t0 = . Then the reliability of x0 ≥ α where α is Cronbach’s alpha.
Here we view the xj as the measured values, the tj as the true values and the ej as the measurement error values. Click here for a proof of Property 1.
Observation: Cronbach’s alpha provides a useful lower bound on reliability (as seen in Property 1). Cronbach’s alpha will generally increase when the correlations between the items increase. For this reason the coefficient measures the internal consistency of the test. Its maximum value is 1, and usually its minimum is 0, although it can be negative (see below).
A commonly-accepted rule of thumb is that an alpha of 0.7 (some say 0.6) indicates acceptable reliability and 0.8 or higher indicates good reliability. Very high reliability (0.95 or higher) is not necessarily desirable, as this indicates that the items may be entirely redundant. These are only guidelines and the actual value of Cronbach’s alpha will depend on many things. E.g. as the number of items increases, Cronbach’s alpha tends to increase too even without any increase in internal consistency.
The goal in designing a reliable instrument is for scores on similar items to be related (internally consistent), but for each to contribute some unique information as well.
Observation: There are an number reasons why Cronbach’s alpha could be low or even negative even for a perfectly valid test. Two such reasons are reverse coding and multiple factors.
Reverse coding: Suppose you use a Likert scale of 1 to 7 with 1 meaning strongly disagree and 7 meaning strongly agree. Suppose two of your questions are: Q1: “I like pizza” and Q20: “I dislike pizza”. These questions ask the same thing, but with reverse wording. In order to apply Cronbach’s alpha properly you need to reverse the scoring of any negatively phrased question, Q20 in our example. Thus if a response to Q20 is say 2, it needs to be scored as 6 instead of 2 (i.e. 8 minus the recorded score).
Multiple factors: Cronbach’s alpha is useful where all the questions are testing more or less the same thing, called a “factor”. If there are multiple factors then you need to determine which questions are testing which factors. If say there are 3 factors (e.g. happiness with your job, happiness with your marriage and happiness with yourself), then you need to split the questionnaire/test into three tests, one containing the questions testing factor 1, one with the questions testing factor 2 and the third with questions testing factor 3. You then calculate Cronbach’s alpha for each of the three tests. The process of determining these “hidden” factors and splitting the test by factor is called Factor Analysis (see Factor Analysis).
Example 1: Calculate Cronbach’s alpha for the data in Example 1 of Kuder and Richardson Formula 20 (repeated in Figure 1 below).
The worksheet in Figure 1 is very similar to the worksheet in Figure 1 of Kuder and Richardson Formula 20. Row 17 contains the variance for each of the questions. E.g. the variance for question 1 (cell B17) is calculated by the formula =VARP(B4:B15). Other key formulas used to calculate Cronbach’s alpha in Figure 1 are described in Figure 2.
Figure 2 – Key formulas for the worksheet in Figure 1
Since the questions only have two answers, Cronbach’s alpha .73082 We see that this is the same as the We see that this is the same as the KR20 reliability calculated for Example 1 of Kuder and Richardson Formula 20.
Observation: If the variances of the xj vary widely, the xj can be standardized to obtain a standard deviation of 1 prior to calculating Cronbach’s alpha.
Observation: To determine how each question on a test impacts the reliability, Cronbach’s alpha can be calculated after deleting the ith variable, for each i ≤ k. Thus for a test with k questions, each with score xj, Cronbach’s alpha is calculated for for all i where = .
If the reliability coefficient increases after an item is deleted, you can assume that the item is not highly correlated with the other items. Conversely, if the reliability coefficient decreases, you can assume that the item is highly correlated with the other items.
Example 2: Calculate Cronbach’s alpha for the survey in Example 1, where any one question is removed.
The necessary calculations are displayed in Figure 3.
Each of the columns B through L represents the test with one question removed. Column B corresponds to question #1, column C corresponds to question #2, etc. Figure 4 displays the formulas corresponding to question #1 (i.e. column B); the formulas for the other questions are similar. Some of the references are to cells shown in Figure 2.
As can be seen from Figure 3, the omission of any single question doesn’t change the Cronbach’s alpha very much. Removal of Q8 affects the result the most.
Observation: Another way to calculate Cronbach’s alpha is to use the Two Factor ANOVA without Replication data analysis tool on the raw data and note that:
Example 3: Calculate the Cronbach’s alpha for Example 1 using ANOVA.
We begin by running Excel’s Anova: Two Factor without Replication data analysis tool using the data in range B4:L15 of the worksheet shown in Figure 1.
As you can see from Figure 5, Cronbach’s alpha is .73802, the same value calculated in Figure 1.
Observation: Alternatively, we could use the Real Statistics Two Factor ANOVA data analysis tool, setting the Number of Rows per Sample to 1. We can also obtain the same result using the following supplemental function.
Real Statistics Functions: The following functions are provided in the Real Statistics Resource Pack:
CRONALPHA(R1, k) = Cronbach’s alpha for the data in range R1 if k = 0 (default) and Cronbach’s alpha with kth item (i.e. column) removed if k > 0
CALPHA(R1): array function which returns a row of Cronbach’s alpha for R1 with each item removed
Thus for the data in Example 1, we can obtain the results shown in Figure 1 and 3 using CRONALPHA(B4:L15) = .738019 and CRONALPHA(B4:L15, 1) = .76321. Also the formula CALPHA(B4:L15) can be used to produce the results shown in range B43:L43 of Figure 3.
Example 4: Calculate Cronbach’s alpha for a 10 question questionnaire with Likert scores between 1 and 7 based on the 15 person sample shown in Figure 6.
As you can see from Figure 6, Cronbach’s alpha is 0.59172, a little below the generally acceptable range. We get the same answer by using the supplemental formula in the Real Statistics Resource Pack, namely CRONALPHA(B4:K18) = 0.59172.
Real Statistics Data Analysis Tool: The Real Statistics Resource Pack provides the Reliability data analysis tool which can be used to calculate Cronbach’s alpha.
We now show how to calculate Cronbach’s alpha for Example 4 using the Reliability data analysis tool. Start, as usual, by pressing Ctrl-m and choose the Reliability option from the menu that appears. Fill in the dialog box that appears as shown in Figure 7.
Figure 7 – Dialog box for Reliability data analysis tool
The output is shown in Figure 8. Cronbach’s alpha is shown in cell M3, while the Cronbach’s alpha values with one question removed are shown in range M8:V8, which is the same as the output from =CALPHA(B4:K18). Note too that the split-half measures are also shown in N12 and N13.
Figure 8 – Cronbach’s alpha option of Reliability data analysis tool