We bring together the basics of variance, standard deviation, correlation coefficients, and hypothesis testing.
Example ① (Alternative Formula for Variance)
Show that the formula variance s² = (mean of x²) - (mean)² can also be used.
Data: 1, 3, 5. Mean = 3. Mean of x² = (1+9+25)/3 = 35/3.
Answer:35/3 - 9 = 35/3 - 27/3 = 8/3. Mean of deviation² = (4+0+4)/3 = 8/3 ✓
Show that the formula variance s² = (mean of x²) - (mean)² can also be used.
Data: 1, 3, 5. Mean = 3. Mean of x² = (1+9+25)/3 = 35/3.
Answer:35/3 - 9 = 35/3 - 27/3 = 8/3. Mean of deviation² = (4+0+4)/3 = 8/3 ✓
Key Point
- Alternative formula for variance: s² = (mean of x²) - (x̄)² (faster calculation)
- Correlation ≠ causation: Causation cannot be stated directly from a correlation
Practice Problems
- Using the formula “mean of x² - mean²,” find the variance of the data: 2, 4, 6.
Answers and Explanations
- Answer: Variance = 8/3 ≒ 2.67. Mean = 4. Mean of x² = (4+16+36)/3 = 56/3. 56/3 - 16 = 56/3 - 48/3 = 8/3.