Population vs. Sample Variance
Variance measures the average squared deviation from the mean, with population variance σ² using n and sample variance s² using (n-1) as denominators — a foundational statistics concept in enVision Algebra 1 Chapter 11 for Grade 11. For {2, 4, 6} as a sample with mean 4: s² = [(2-4)² + (4-4)² + (6-4)²]/(3-1) = 8/2 = 4. For population {10, 12, 14, 16} with mean 13: σ² = [(10-13)²+(12-13)²+(14-13)²+(16-13)²]/4 = (9+1+1+9)/4 = 5. Variance is the square of standard deviation and represents spread in squared units.
Key Concepts
Variance is the average of the squared differences from the mean:.
Sample variance: $s^2 = \frac{\sum(x \bar{x})^2}{n 1}$.
Common Questions
What is the difference between sample and population variance formulas?
Population variance: σ² = Σ(x-μ)²/n. Sample variance: s² = Σ(x-x̄)²/(n-1). The only difference is n vs. n-1 in the denominator; n-1 corrects for estimation bias.
Calculate the sample variance for {2, 4, 6}.
Mean x̄ = 4. Deviations: (2-4)² = 4, (4-4)² = 0, (6-4)² = 4. Sum = 8. s² = 8/(3-1) = 8/2 = 4.
Calculate the population variance for {10, 12, 14, 16}.
Mean μ = 13. Deviations: 9, 1, 1, 9. Sum = 20. σ² = 20/4 = 5.
How does variance relate to standard deviation?
Standard deviation is the square root of variance: σ = √σ² for populations and s = √s² for samples. Variance is in squared units; standard deviation is in the original units.
Why is variance useful even though it uses squared units?
Squaring the deviations eliminates negative values, ensuring negative and positive deviations don't cancel. It also emphasizes large deviations more than small ones. Standard deviation converts back to original units for easier interpretation.