A stock had returns of 5, 14, 11, -8, and 6 percent over the past 5 years. The standard deviation for this stock is:

Question

BlackTom AI · Accepted Answer

Standard deviation measures dispersion of the returns around their mean. For a small set of historical returns we use the sample standard deviation (divide by n–1) to estimate variability.

1) Relevant knowledge points and concepts
- Compute the arithmetic mean of the returns.
- Find each return’s deviation from the mean, square those deviations, sum them.
- For a sample of n observations use the sample variance = (sum of squared deviations)/(n−1).
- Sample standard deviation = square root of the sample variance. Results are expressed in the same units (percent).

2) Step-by-step reasoning and 3) intermediate calculations
- Returns: 5%, 14%, 11%, −8%, 6%. n = 5.
- Mean = (5 + 14 + 11 + (−8) + 6) / 5 = 28 / 5 = 5.6%.
- Deviations from mean:
  - 5 − 5.6 = −0.6
  - 14 − 5.6 = 8.4
  - 11 − 5.6 = 5.4
  - −8 − 5.6 = −13.6
  - 6 − 5.6 = 0.4
- Squared deviations:
  - (−0.6)^2 = 0.36
  - 8.4^2 = 70.56
  - 5.4^2 = 29.16
  - (−13.6)^2 = 184.96
  - 0.4^2 = 0.16
- Sum of squared deviations = 0.36 + 70.56 + 29.16 + 184.96 + 0.16 = 285.20
- Sample variance = 285.20 / (n − 1) = 285.20 / 4 = 71.30
- Sample standard deviation = sqrt(71.30) = 8.444...%

4) Why the correct answer is correct
Dividing by n−1 (4) is the standard approach when these five years are treated as a sample of possible returns; that yields variance = 71.30 and standard deviation ≈ 8.444%, which rounds to 8.44%.

5) Why other options are incorrect
- 7.74% and 7.49%: these values arise from incorrect division or arithmetic errors (e.g., using an incorrect mean or incorrect denominator). They do not match the correct sample calculation.
- 9.68%: too large; would result from an inflated variance (arithmetic mistake or including an extra term). It is not produced by the correct squared-deviation sum and n−1 division.

6) Conclusion
The computed sample standard deviation is approximately 8.444%, rounded to 8.44%, which matches the provided correct answer 8.44%.

A stock had returns of 5, 14, 11, -8, and 6 percent over the past 5 years. The standard deviation for this stock is:单项选择题

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