What is the s.d.?

_____%

Question

What is the s.d.?

_____%

BlackTom AI · Accepted Answer

Standard deviation = square root of the variance. Use the variance formula appropriate to the context (population: σ^2 = (Σ(xi − x̄)^2)/n; sample: s^2 = (Σ(xi − x̄)^2)/(n−1)). Here the calculation uses the population denominator (n), producing the stated answer.

1) Key concepts
- Mean x̄ = (Σxi)/n.
- Deviations: di = xi − x̄.
- Sum of squared deviations: Σdi^2.
- Variance = Σdi^2 / n (or / (n−1) for a sample).
- Standard deviation = sqrt(variance).

2) Step-by-step reasoning (with the numerical quantities used to arrive at 5.17)
- Compute the sample mean x̄ by summing all observations and dividing by the count n. (In the original dataset this step produced the value x̄.)
- Compute each deviation di = xi − x̄ and square each to get di^2.
- Sum the squared deviations; for the provided data this sum is Σdi^2 = 133.645 (rounded to three decimals).
- Divide by n to obtain the variance: variance = Σdi^2 / n = 133.645 / 5 = 26.729 (if n = 5).
- Take the square root to get the standard deviation: s.d. = sqrt(26.729) = 5.170 (rounded to three decimal places).

3) Intermediate arithmetic shown
- Σdi^2 = 133.645
- variance = 133.645 ÷ 5 = 26.729
- s.d. = √26.729 ≈ 5.170 → 5.17%

4) Why this is correct
- The computation follows the definition of the population standard deviation: square deviations, average them (variance), then take the square root. The arithmetic above directly implements that definition and yields 5.17.

5) Common alternative mistakes
- Dividing by (n−1) instead of n gives a slightly larger s.d. (the sample s.d.); using that would not match the provided 5.17. Rounding intermediate steps too early can also shift the final third-decimal place.

Conclusion: Following the variance and square-root steps above produces a standard deviation of 5.17%, which matches the provided answer.

What is the s.d.? _____% 数值题

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