R3 练习: 统计度量
考纲范围
- Calculate, interpret, and evaluate measures of central tendency and location to address an investment problem.
- Calculate, interpret, and evaluate measures of dispersion to address an investment problem.
- Interpret and evaluate measures of skewness and kurtosis to address an investment problem.
- Interpret correlation between two variables to address an investment problem.
Q1.
Examine the annual returns from 2018 to 2022, depicted in the following table for two funds, namely Fund A and Fund B.
| 2018 | 2019 | 2021 | 2022 | |
|---|---|---|---|---|
| Fund A | -12% | 5% | 10% | 4% |
| Fund B | 3% | -4% | 7% | 7% |
Which of the following statements is correct?
A. The median annual return for Fund A is 7.5%.
B. The arithmetic mean annual return for Fund A is higher than its median annual return.
C. The mode for Fund B is 7%.
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答案：C
解析：逐一验证各选项。
计算过程：
- Fund A排序：-12%, 4%, 5%, 10%。中位数 = (4% + 5%)/2 = 4.5%
- Fund A算术平均 = (-12 + 5 + 10 + 4)/4 = 7/4 = 1.75%
- Fund B的众数：3%, -4%, 7%, 7%。7%出现两次，是众数。
选项 判断 解析 A ✗ Fund A中位数 = (4%+5%)/2 = 4.5%，不是7.5% B ✗ Fund A算术平均1.75% < 中位数4.5%，因为存在极端负值-12%拉低了均值 C ✓ Fund B中7%出现两次，是唯一重复值，因此众数为7%
Q2.
Which of the following statements is least likely correct regarding mode?
A. When having a single value that is most frequently occurring, the distribution is called unimodal.
B. Trimodal happens when the distribution has three most frequently occurring values.
C. A distribution must have at least one mode.
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答案：C
解析：众数（mode）是数据集中出现频率最高的值。
选项 判断 解析 A ✓ 正确，只有一个最高频率值时称为单峰分布（unimodal） B ✓ 正确，三个最高频率值时称为三峰分布（trimodal） C ✗ 不正确。当所有值出现次数相同时，分布没有众数（no mode）。因此分布不一定有众数
Q3.
The value of the first quintile of a data set {1, 9, 6, 18, 47, 25} is closest to:
A. 2.
B. 3.
C. 8.
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答案：B
解析：先排序，再计算第一个五分位数（quintile）的位置。
计算过程：
- 排序后：{1, 6, 9, 18, 25, 47}，共n=6个数据
- 第一个五分位数对应的百分位数 = 20th percentile
- 位置 L = (n+1) × (y/100) = 7 × 0.20 = 1.4
- 即第1个值与第2个值之间，偏移0.4
- Q₁₅ = X₁ + 0.4 × (X₂ - X₁) = 1 + 0.4 × (6 - 1) = 1 + 2 = 3
选项 判断 解析 A ✗ 2可能是位置值的误算 B ✓ 第20百分位 = 1 + 0.4 × (6-1) = 3 C ✗ 8可能混淆了分位数的位置
Q4.
Which of the following statements about quantile is most accurate?
A. Quantiles are often used to rank performance and investment research.
B. For ascending ordered observations, the value of the 4th quintile is equal to the value of the 3rd quartile.
C. For ascending ordered observations, the value of the 60th percentile is equal to (number of data +1)*0.6.
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答案：A
解析：分位数在投资分析中的应用和性质。
选项 判断 解析 A ✓ 正确，分位数常用于排名绩效和投资研究 B ✗ 第4个五分位数 = 80th percentile，第3个四分位数 = 75th percentile，两者不相等 C ✗ (n+1)×0.6是第60百分位数的位置（location），不是其值（value）。值需要根据位置在数据中进行插值
Q5.
A box and whisker plot for a set of data is shown below:
- Maximum: 12.30
- Q3: 9.78
- Median: 8.15
- Mean: 7.38 (marked with ×)
- Q1: 4.80
- Minimum: 1.35
The interquartile range (IQR) is closest to:
A. 8.15.
B. 7.38.
C. 4.98.
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答案：C
解析：四分位距（IQR）是第三四分位数与第一四分位数的差。
计算过程：
选项 判断 解析 A ✗ 8.15是中位数，不是IQR B ✗ 7.38是均值，不是IQR C ✓ IQR = Q₃ - Q₁ = 9.78 - 4.80 = 4.98
Q6.
In terms of the measures of dispersion for a distribution, which of the following statements is least likely to be accurate?
A. The range is defined as the difference between maximum value and minimum value.
B. The arithmetic average of the deviations around the mean must be equal to zero.
C. Standard deviation will be either less than or equal to the corresponding variance.
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答案：C
解析：考察离散程度度量的概念。
选项 判断 解析 A ✓ 正确，极差 = 最大值 - 最小值 B ✓ 正确，各偏差之和（及其算术平均）恒等于零，这是均值的性质 C ✗ 不正确。当方差 < 1时，标准差 > 方差（如方差=0.25，标准差=0.5）；当方差 > 1时，标准差 < 方差。因此不能笼统地说标准差总是小于或等于方差
Q7.
A mutual fund’s annual returns over the past 8 years are presented as follows:
| Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Return (%) | 4.8 | 5.5 | 1.5 | -3.0 | 0.2 | 4 | 3.2 | 1.2 |
Assume the target annual return is 1.5%, and the target downside deviation of the returns over the past 8 years is closest to:
A. 1.77%.
B. 1.87%.
C. 1.97%.
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答案：A
解析：目标下行偏差（target downside deviation）只考虑低于目标收益率的偏差。
计算过程： 目标收益率 = 1.5%。低于目标的年份：
- Year 4: -3.0% → (-3.0 - 1.5)² = (-4.5)² = 20.25
- Year 5: 0.2% → (0.2 - 1.5)² = (-1.3)² = 1.69
- Year 8: 1.2% → (1.2 - 1.5)² = (-0.3)² = 0.09
注意：高于目标的年份偏差计为0。
但如果使用n-1（样本）：
选项 判断 解析 A ✓ 使用n-1=7计算的目标下行偏差 ≈ 1.77% B ✗ 计算偏差 C ✗ 计算偏差
Q8.
Stock A has an average return of 13% and a standard deviation of 3%. Stock B has an average return of 30% and a standard deviation of 10%. Given a risk-free rate of 5%, which investment is more attractive based on the coefficient of variation?
A. Stock B.
B. Stock A.
C. Either is ok.
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答案：B
解析：变异系数（CV）衡量每单位收益承担的风险，CV越小，风险收益比越好。
计算过程：
CV_A < CV_B，说明Stock A每单位收益承担的风险更低，因此Stock A更具吸引力。
选项 判断 解析 A ✗ Stock B的CV更高（0.33 > 0.23），单位收益风险更大 B ✓ Stock A的CV更低，单位收益所承受的风险更小，更有吸引力 C ✗ 两只股票的CV不同，不是无差异的
Q9.
With respect to skewness, which of the following statements is most accurate?
A. The calculation of skewness is related to the third power.
B. A positively skewed return distribution has frequent extreme gains.
C. Investors usually prefer negatively skewed return distribution.
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答案：A
解析：偏度（skewness）的计算和特征。
选项 判断 解析 A ✓ 正确，偏度的计算涉及偏差的三次方（third power/moment） B ✗ 正偏分布有少量的极端大值收益（infrequent extreme gains），不是频繁的 C ✗ 投资者通常偏好正偏分布（frequent small losses, few extreme gains），不偏好负偏分布
Q10.
If a return distribution tends to get a few great losses with more frequent small gains, then:
A. this distribution is negatively skewed.
B. this distribution’s mean is larger than its median.
C. this distribution is symmetrical.
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答案：A
解析：少量大额亏损伴随频繁的小额盈利，这是负偏分布的典型特征。
选项 判断 解析 A ✓ 少量极端损失+频繁小额收益 = 负偏分布（左尾较长） B ✗ 在负偏分布中，均值 < 中位数 < 众数，均值被极端负值拉低 C ✗ 对称分布不会有偏向一侧的极端值
Q11.
Two distributions, A and B, have the same mode. If distribution A is positively skewed and distribution B is symmetrical, then compared with B, the median of A is:
A. the same.
B. bigger.
C. smaller.
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答案：B
解析：对于对称分布B：均值 = 中位数 = 众数。对于正偏分布A：众数 < 中位数 < 均值。
既然两个分布的众数相同，而A是正偏的（中位数 > 众数），B是对称的（中位数 = 众数），因此A的中位数 > B的中位数。
选项 判断 解析 A ✗ 相同众数下，正偏分布的中位数大于对称分布的中位数 B ✓ 正偏分布中：中位数 > 众数 = B的中位数，所以A的中位数更大 C ✗ 方向反了
Q12.
Which of the following statements regarding kurtosis is most accurate?
A. Kurtosis of a normal distribution is zero.
B. Kurtosis indicates whether a distribution is symmetrical about its mean.
C. Kurtosis tells us whether a distribution has fatter tails than a normal distribution.
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答案：C
解析：峰度（kurtosis）衡量分布尾部的厚度和峰值的尖锐程度。
选项 判断 解析 A ✗ 正态分布的峰度（kurtosis）= 3，超额峰度（excess kurtosis）= 0。题目说的是kurtosis而非excess kurtosis B ✗ 判断分布是否对称的是偏度（skewness），不是峰度 C ✓ 峰度告诉我们分布是否具有比正态分布更厚的尾部（尖峰厚尾或平峰薄尾）
Q13.
If the returns on two assets tend to be above or below their expected values simultaneously, the covariance of these two assets’ returns is:
A. positive.
B. negative.
C. zero.
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答案：A
解析：协方差衡量两个变量同方向变动的程度。当两个资产同时高于或低于各自的期望值时，偏差的乘积为正，协方差为正。
选项 判断 解析 A ✓ 同向变动（同时高于或低于期望值）→ 正协方差 B ✗ 反向变动才产生负协方差 C ✗ 零协方差表示线性无关
Q14.
The research shows that the correlation coefficient between the return of NASDAQ and S&P 500 is 0.90. The variance of returns on NASDAQ is 0.0016, and the variance of returns on S&P 500 is 0.0025. What is the covariance of returns on these two indexes?
A. 0.000036.
B. 0.0058.
C. 0.0018.
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答案：C
解析：利用相关系数和标准差计算协方差。
计算过程：
选项 判断 解析 A ✗ 0.000036 = 0.0016 × 0.0025 × 0.9，错误地用方差代替标准差相乘 B ✗ 计算错误 C ✓ Cov = 0.90 × 0.04 × 0.05 = 0.0018
Q15.
If two variables have a strong linear relationship, the correlation coefficient between these two variables is most likely to be:
A. -0.9.
B. 0.
C. 0.5.
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答案：A
解析：相关系数的绝对值越接近1，线性关系越强。“Strong linear relationship”指的是|ρ|接近1。
选项 判断 解析 A ✓ B ✗ 0表示无线性关系 C ✗
Q16.
Which of the following is not a limitation of correlation analysis?
A. Correlation does not indicate causation but simply association.
B. The correlation is quite sensitive to outliers which may lead to unreliable correlation.
C. The correlation coefficients measure linear relationships which are statistics with no unit that can be compared directly.
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答案：C
解析：需要识别哪个不是相关分析的局限性，而是相关分析的优点。
选项 判断 解析 A 是局限 相关不代表因果，这是相关分析的著名局限性 B 是局限 相关系数对离群值敏感，可能导致不可靠的结果 C 不是局限 相关系数无量纲且可以直接比较，这是相关系数的优点而非局限