Cronbach's Alpha: Complete Step-by-Step Solution
1 · Total Score Calculation
| ID | Q1 | Q2 | Q3 | Q4 | Q5 | Total |
|---|
| 1 | 5 | 5 | 5 | 5 | 5 | 25 |
| 2 | 4 | 4 | 5 | 4 | 5 | 22 |
| 3 | 5 | 5 | 5 | 5 | 5 | 25 |
| 4 | 4 | 4 | 4 | 5 | 5 | 22 |
| 5 | 4 | 4 | 5 | 5 | 5 | 23 |
| 6 | 4 | 4 | 5 | 5 | 5 | 23 |
| 7 | 5 | 5 | 5 | 5 | 5 | 25 |
| 8 | 4 | 4 | 4 | 4 | 4 | 20 |
| 9 | 5 | 5 | 5 | 5 | 5 | 25 |
| 10 | 4 | 5 | 5 | 4 | 5 | 23 |
| 11 | 5 | 5 | 5 | 4 | 4 | 23 |
| 12 | 5 | 5 | 5 | 5 | 5 | 25 |
| 13 | 5 | 5 | 5 | 5 | 5 | 25 |
| 14 | 5 | 5 | 5 | 4 | 5 | 24 |
| 15 | 4 | 4 | 4 | 4 | 4 | 20 |
| 16 | 5 | 4 | 4 | 4 | 4 | 21 |
| 17 | 3 | 4 | 5 | 4 | 3 | 19 |
| 18 | 4 | 3 | 3 | 3 | 3 | 16 |
| 19 | 4 | 4 | 3 | 5 | 5 | 21 |
| 20 | 5 | 5 | 4 | 5 | 5 | 24 |
| 21 | 5 | 5 | 4 | 3 | 4 | 21 |
| 22 | 3 | 5 | 5 | 5 | 4 | 22 |
| 23 | 3 | 4 | 5 | 5 | 5 | 22 |
| 24 | 3 | 4 | 5 | 4 | 5 | 21 |
| 25 | 4 | 5 | 5 | 4 | 5 | 23 |
| 26 | 5 | 4 | 4 | 4 | 4 | 21 |
| 27 | 4 | 4 | 4 | 4 | 4 | 20 |
| 28 | 5 | 4 | 5 | 5 | 5 | 24 |
| 29 | 4 | 5 | 5 | 5 | 5 | 24 |
| 30 | 5 | 5 | 5 | 5 | 4 | 24 |
| Σ | 130 | 134 | 138 | 134 | 136 | 672 |
Total Sum = 672 | Mean Total Score:
\(\displaystyle \bar{x}_{\text{total}} = \frac{672}{30} = \mathbf{22.4}\)
2 · Sample Variance of Each Question
Formula:
\(\displaystyle s^2 = \frac{\displaystyle\sum x_i^2 - \dfrac{(\sum x_i)^2}{n}}{n - 1}\)
Question 1 (Q1)
- \(\sum x_i = 130\)
- \(\sum x_i^2 = 5^2(14) + 4^2(12) + 3^2(4)\) \(= 350 + 192 + 36 = 578\)
- \(\frac{(\sum x_i)^2}{n} = \frac{130^2}{30} = 563.3333\)
- \(SS = 578 - 563.3333 = 14.6667\)
\(s^2_{Q1} = \dfrac{14.6667}{29} = \mathbf{0.5057}\)
Question 2 (Q2)
- \(\sum x_i = 134\)
- \(\sum x_i^2 = 5^2(15) + 4^2(14) + 3^2(1)\) \(= 375 + 224 + 9 = 608\)
- \(\frac{(\sum x_i)^2}{n} = \frac{134^2}{30} = 598.5333\)
- \(SS = 608 - 598.5333 = 9.4667\)
\(s^2_{Q2} = \dfrac{9.4667}{29} = \mathbf{0.3264}\)
Question 3 (Q3)
- \(\sum x_i = 138\)
- \(\sum x_i^2 = 5^2(20) + 4^2(8) + 3^2(2)\) \(= 500 + 128 + 18 = 646\)
- \(\frac{(\sum x_i)^2}{n} = \frac{138^2}{30} = 634.8\)
- \(SS = 646 - 634.8 = 11.2\)
\(s^2_{Q3} = \dfrac{11.2}{29} = \mathbf{0.3862}\)
Question 4 (Q4)
- \(\sum x_i = 134\)
- \(\sum x_i^2 = 5^2(16) + 4^2(12) + 3^2(2)\) \(= 400 + 192 + 18 = \mathbf{610}\)
- \(\frac{(\sum x_i)^2}{n} = \frac{134^2}{30} = 598.5333\)
- \(SS = 610 - 598.5333 = 11.4667\)
\(s^2_{Q4} = \dfrac{11.4667}{29} = \mathbf{0.3954}\)
Question 5 (Q5)
- \(\sum x_i = 136\)
- \(\sum x_i^2 = 5^2(18) + 4^2(10) + 3^2(2)\) \(= 450 + 160 + 18 = 628\)
- \(\frac{(\sum x_i)^2}{n} = \frac{136^2}{30} = 616.5333\)
- \(SS = 628 - 616.5333 = 11.4667\)
\(s^2_{Q5} = \dfrac{11.4667}{29} = \mathbf{0.3954}\)
Sum of Item Variances
\(\displaystyle \sum \sigma^2_i = 0.5057 + 0.3264 + 0.3862 + 0.3954 + 0.3954 = \mathbf{2.0090}\)
3 · Variance of Total Scores
| Total | Frequency | Total² | Freq × Total² |
|---|
| 25 | 6 | 625 | 3,750 |
| 24 | 5 | 576 | 2,880 |
| 23 | 4 | 529 | 2,116 |
| 22 | 5 | 484 | 2,420 |
| 21 | 5 | 441 | 2,205 |
| 20 | 3 | 400 | 1,200 |
| 19 | 1 | 361 | 361 |
| 16 | 1 | 256 | 256 |
| Σ | 30 | - | 15,188 |
- \(\sum \text{Total}_i^2 = 15{,}188\)
- \(\frac{672^2}{30} = \frac{451{,}584}{30}\) \(= 15{,}052.8\)
- \(SS = 15{,}188 - 15{,}052.8 = 135.2\)
\(s^2_{\text{total}} = \dfrac{135.2}{29} = \mathbf{4.6621}\)
4 · Cronbach's Alpha Calculation
\(\displaystyle \alpha = \frac{k}{k - 1} \left(1 - \frac{\displaystyle\sum \sigma^2_i}{\sigma^2_{\text{total}}}\right)\)
Where: \(k = 5\), \(\sum \sigma^2_i = 2.0090\), \(\sigma^2_{\text{total}} = 4.6621\)
- 1
\(\displaystyle\frac{2.0090}{4.6621} = 0.4309\)
- 2
\(1 - 0.4309 = 0.5691\)
- 3
\(\dfrac{k}{k-1} = \dfrac{5}{4} = 1.25\)
- 4
\(\alpha = 1.25 \times 0.5691 = 0.7114\)
\(\boxed{\alpha = \mathbf{0.7114}}\)
5 · Interpretation
Cronbach's Alpha = 0.7114 - Exceeds the 0.70 threshold, indicating acceptable internal consistency reliability. The 5 items consistently measure the same construct.
Suggested write-up:
"The internal consistency of the instrument was assessed using Cronbach's alpha (α = 0.71), which exceeds the minimum acceptable value of 0.70 (Nunnally & Bernstein, 1994), indicating that the scale demonstrates acceptable reliability."
Citations:
Nunnally, J. C., & Bernstein, I. H. (1994). Psychometric theory (3rd ed.). McGraw-Hill.
Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297-334.
Final Summary
\(\sum \sigma^2_i\)
2.0090
\(s^2_{\text{total}}\)
4.6621
✅ Acceptable reliability - α ≥ 0.70