位置24的问题 A stock had returns of 5 percent, 14 percent, 11 percent, −8 percent, and 6 percent over the past five years. What is the standard deviation of these returns?7.74%8.44%9.68%7.49%8.21%清除选择

Question

BlackTom AI · Accepted Answer

将问题中的五个年化回报率列出并逐一分析，便于比较。
数据为：5%, 14%, 11%, -8%, 6%。
第一步，计算平均值（样本均值）以衡量数据的中心趋势。将数值相加得到总和：5 + 14 + 11 - 8 + 6 = 28，五个样本的均值为 28 / 5 = 5.6%。
接下来计算每个观察值相对均值的偏差，然后平方并求和以得到方差分子。
- (5 - 5.6)² = (-0.6)² = 0.36
- (14 - 5.6)² = 8.4² = 70.56
- (11 - 5.6)² = 5.4² = 29.16
- (-8 - 5.6)² = (-13.6)² = 184.96
- (6 - 5.6)² = 0.4² = 0.16
将这些平方偏差相加：0.36 + 70.56 + 29.16 + 184.96 + 0.16 = 285.2。
随后区分两种方差的定义：
- 若使用总体方差（n 分母），方差 = 285.2 / 5 = 57.04，标准差 = sqrt(57.04) ≈ 7.55%，与选项不完全吻合。
- 若使用样本方差（n-1 分母），方差 = 285.2 / 4 = 71.3，标准差 = sqrt(71.3) ≈ 8.44%，这与给出的选项之一 8.44% 相匹配。
在金融数据中，经常以样本标准差来衡量收益率的波动性，因此选择 8.44% 作为对应的标准差。
现在逐一评估选项，帮助理解为何其它值不合适。
选项 7.74%：这个值接近合理的误差区间，但与基于样本分母的计算结果 8.44% 相去较远，因此不符合基于样本标准差的标准计算。
选项 8.44%：这是基于样本标准差的正确计算结果，反映了数据在均值周围的实际离散程度。
选项 9.68%：比 8.44% 稍大，若按样本方差去计算，需要更高的偏差量，然而在此组数据的平方偏差总和下并非对应的值。
选项 7.49%：略低于 7.55% 的近似值，未能匹配通过样本方差得到的结果，因此不正确。
选项 8.21%：也远离 8.44% 的结果，可能来自对均值或偏差计算的不同处理，但并非该数据集合的标准样本差异值。

位置24的问题 A stock had returns of 5 percent, 14 percent, 11 percent, −8 percent, and 6 percent over the past five years. What is the standard deviation of these returns?7.74%8.44%9.68%7.49%8.21%清除选择

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