Answer: A more complex formula involving d₁² and d₂² (distances from combined mean).
- A (n₁σ₁²+n₂σ₂²)/(n₁+n₂), which overlooks the shift in means between the two groups
- B A more complex formula involving d₁² and d₂² (distances from combined mean)
- C (σ₁²+σ₂²)/2, a simple unweighted average that ignores group sizes
- D σ₁ × σ₂, the simple product of the two individual standard deviations
Correct answer: B. A more complex formula involving d₁² and d₂² (distances from combined mean)
Explanation: Combined mean x̄ = (n₁x̄₁+n₂x̄₂)/(n₁+n₂). Let d₁ = x̄₁−x̄ and d₂ = x̄₂−x̄. Combined variance = [n₁(σ₁²+d₁²) + n₂(σ₂²+d₂²)]/(n₁+n₂). The d² terms capture the shift between group means and the combined mean.
In a perfectly normal (bell-shaped) distribution, mean, median, and mode all coincide at the centre; about 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 - the empirical rule used to judge how typical or extreme a value is.
Concept context
Mean, median, mode, standard deviation, and data interpretation