Two samples have the same mean test score. One sample has a much larger standard deviation than the other. What does this indicate?

Question

BlackTom AI · Accepted Answer

A larger standard deviation means the observations are more spread out around the mean, so the correct interpretation is: Scores in the sample with the larger SD are more spread out.

1) Relevant concepts
- Mean (x̄) measures the central tendency — the average value.
- Standard deviation (SD) measures the typical distance of individual scores from the mean. For a population: SD = sqrt[ (1/N) Σ (xi − x̄)² ]. For a sample, a similar formula uses (n−1) in the denominator for an unbiased estimate.
- Variance is SD squared and equals the average of the squared deviations from the mean.

2) Step-by-step reasoning
- Same mean for both samples means their centers are equal.
- SD compares the average squared deviations from that shared center.
- A larger SD implies the average squared deviation (and therefore average absolute deviation) is larger, so individual scores lie farther from the mean on average.

3) Intermediate/calculation illustration
- Example: Both samples have mean 70.
  Sample A: scores 68, 70, 72 → deviations −2,0,2 → squared 4,0,4 → variance = (4+0+4)/3 = 8/3 ≈ 2.67 → SD ≈ 1.63.
  Sample B: scores 55, 70, 85 → deviations −15,0,15 → squared 225,0,225 → variance = (225+0+225)/3 = 150 → SD ≈ 12.25.
- Sample B’s SD is much larger, showing its scores are much more spread out from the mean.

4) Why the correct answer is correct
- By definition SD quantifies spread about the mean. Higher SD = greater average distance from the mean, so scores are more dispersed.

5) Why other interpretations are incorrect
- It does not mean the mean is different (we are told means are the same).
- It doesn’t necessarily mean more high scores or more low scores; it means greater variability in both directions.
- While outliers can increase SD, a larger SD does not automatically prove outliers are present—just greater overall dispersion.
- SD is related to variability, not sample size (though sample size affects the precision of the SD estimate).

6) Conclusion
This confirms the provided answer: Scores in the sample with the larger SD are more spread out.

Two samples have the same mean test score. One sample has a much larger standard deviation than the other. What does this indicate?Single choice

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