How is cronbach alpha coefficient calculated

Cronbach's alpha is a measure of internal consistency or reliability of a set of scale or test items. It is commonly used in psychology, education, and various social sciences to assess the reliability of a survey or questionnaire. The value of Cronbach's alpha ranges from 0 to 1, with higher values indicating greater reliability.

Here's how you can calculate Cronbach's alpha:

### Formula

The formula for Cronbach's alpha (\(\alpha\)) is:

\[
\alpha = \frac{N}{N-1} \left(1 - \frac{\sum s_i^2}{s_t^2}\right)
\]

Where:  
- \(N\) = the number of items (questions) in the scale  
- \(s_t^2\) = the variance of the total scores  
- \(s_i^2\) = the variance of the individual items  

### Steps for Calculation

1. **Data Collection**: Gather the responses for the items in your scale. The data should be organized such that each row corresponds to an individual respondent and each column corresponds to an item.

2. **Calculate Total Scores**: For each respondent, calculate the total score by summing their responses across all items.

3. **Calculate Variances**:
   - Calculate the variance of total scores (\(s_t^2\)).
   - For each item, calculate the variance (\(s_i^2\)).

4. **Sum the Item Variances**: Add the variances of all individual items together.

5. **Substitute into the Formula**: Plug the values into the formula to compute Cronbach's alpha.

### Interpretation

- **\(\alpha \geq 0.9\)**: Excellent reliability
- **0.7 < \(\alpha\) < 0.9**: Good reliability
- **0.6 < \(\alpha\) < 0.7**: Acceptable reliability
- **\(\alpha < 0.6\)**: Poor reliability

### Example Calculation

Suppose you have a simple scale with 3 items (A, B, C) and you collected data from 5 respondents:

|   | A | B | C |
|---|---|---|---|
| 1 | 3 | 4 | 3 |
| 2 | 2 | 3 | 4 |
| 3 | 4 | 4 | 5 |
| 4 | 3 | 2 | 3 |
| 5 | 4 | 4 | 4 |

1. **Total Scores**: 
   - Respondent 1: 10, Respondent 2: 9, Respondent 3: 13, Respondent 4: 8, Respondent 5: 12
2. **Calculate Variances**:
   - Total scores variance \((s_t^2)\): Calculate the variance of [10, 9, 13, 8, 12].
   - Individual item variances \((s_i^2)\): Calculate variances for A, B, and C.
3. **Sum the Item Variances**: Compute \( \sum s_i^2 \).
4. **Apply the Formula**: Substitute these variances into the Cronbach's alpha formula to find the value.

Using statistical software (like R, SPSS, Python, etc.) can simplify the calculation, especially for larger datasets.