📖 描述性统计综合练习 + 周测

Lesson 114: Descriptive Statistics Comprehensive Review + Weekly Quiz

Learning Objectives

By the end of this lesson, candidates should be able to:

1. Summarize and differentiate the key measures of central tendency: arithmetic mean, geometric mean, median, and mode.
2. Calculate and interpret measures of dispersion: range, mean absolute deviation (MAD), variance, and standard deviation.
3. Describe skewness and kurtosis as measures of the shape of a distribution.
4. Apply Chebyshev's inequality to determine the proportion of observations within a given number of standard deviations from the mean.
5. Calculate and interpret the coefficient of variation (CV) and the Sharpe ratio for comparing risk-adjusted returns.
6. Compute and interpret quartiles, quintiles, deciles, and percentiles.
7. Distinguish between population parameters and sample statistics.

1. Comprehensive Topic Review

1.1 Measures of Central Tendency

Central tendency describes the center of a data distribution — the value around which observations cluster.

Arithmetic Mean (¯x or μ)

The arithmetic mean is the sum of all observations divided by the number of observations.

Population mean:

μ = (Σ Xᵢ) / N

Sample mean:

¯x = (Σ Xᵢ) / n

Properties: - Most commonly used measure of central tendency - Sensitive to extreme values (outliers) - Uses all data points

Weighted Mean:

Used when different observations carry different weights:

¯x_w = Σ (wᵢ × Xᵢ) / Σ wᵢ

Portfolio returns are a classic application: each asset's return is weighted by its proportion of the portfolio.

Geometric Mean (G):

The geometric mean is the nth root of the product of n observations. It is the correct measure for calculating average compound growth rates.

G = (X₁ × X₂ × ... × Xₙ)^(1/n)

For returns data, the geometric mean of (1 + Rₜ) minus 1 gives the compound annual growth rate (CAGR):

R_G = [(1+R₁)(1+R₂)...(1+Rₙ)]^(1/n) − 1

Key relationship: The geometric mean ≤ arithmetic mean, with equality only when all observations are identical.

Median:

The midpoint of a sorted data set. For an odd number of observations, it is the middle value; for an even number, it is the average of the two middle values.

• Not affected by outliers
• Preferred for skewed distributions

Mode:

The most frequently occurring value in a data set. A data set can have one mode (unimodal), two modes (bimodal), or more (multimodal). If no value repeats, there is no mode.

1.2 Quantiles

Quantiles divide a data set into equal-sized groups after sorting.

• Quartiles: Divide data into 4 parts (Q1, Q2 = median, Q3)
• Quintiles: Divide data into 5 parts
• Deciles: Divide data into 10 parts
• Percentiles: Divide data into 100 parts

Position of the kth percentile (linear interpolation method):

L_y = (n + 1) × (y / 100)

where y is the desired percentile rank and n is the number of observations. If L_y is not an integer, interpolate between adjacent observations.

1.3 Measures of Dispersion

Dispersion measures the spread or variability of data around the central value.

Range:

Range = Maximum Value − Minimum Value

Simple but uses only two data points; sensitive to outliers.

Mean Absolute Deviation (MAD):

MAD = Σ |Xᵢ − ¯x| / n

The average absolute distance from the mean. More robust than variance as it does not square deviations.

Population Variance (σ²):

σ² = Σ (Xᵢ − μ)² / N

Population Standard Deviation (σ):

σ = √σ²

Sample Variance (s²):

s² = Σ (Xᵢ − ¯x)² / (n − 1)

Note the denominator is (n − 1) — this is the degrees of freedom adjustment (Bessel's correction), which corrects for the bias introduced by estimating the population mean from the sample.

Sample Standard Deviation (s):

s = √s²

Why use (n − 1)? Using the sample mean rather than the true population mean makes the squared deviations systematically smaller. Dividing by (n − 1) instead of n corrects this bias; the resulting s² is an unbiased estimator of σ².

Semi-variance / Semi-deviation:

Measures downside risk by considering only observations below the mean (or below a target). This is relevant for risk-averse investors who care more about negative deviations.

1.4 Chebyshev's Inequality

For any data set (regardless of distribution shape), Chebyshev's inequality states that the proportion of observations within k standard deviations of the mean is at least:

1 − (1 / k²)

This is a conservative bound. For normally distributed data, the actual percentages are much higher (e.g., ~95% within 2σ vs. Chebyshev's 75%).

1.5 Measures of Shape

Skewness describes the symmetry of a distribution.

• Positive skew (right-tailed): Mean > Median > Mode. The right tail is longer; extreme positive values pull the mean upward.
• Negative skew (left-tailed): Mean < Median < Mode. The left tail is longer; extreme negative values pull the mean downward.
• Zero skew (symmetric): Mean ≈ Median ≈ Mode.

Rule of thumb for skewed distributions: The median is the preferred measure of central tendency because it is not distorted by extreme values.

Kurtosis measures the "tailedness" or peakedness of a distribution.

• Mesokurtic: Same kurtosis as the normal distribution (excess kurtosis = 0)
• Leptokurtic: Heavy tails, more peaked (excess kurtosis > 0). Higher probability of extreme outcomes.
• Platykurtic: Light tails, flatter (excess kurtosis < 0). Lower probability of extreme outcomes.

Excess kurtosis = sample kurtosis − 3

Financial significance: Leptokurtic distributions (excess kurtosis > 0) imply more frequent extreme events (tail risk) than a normal distribution would predict. This is critical for risk management.

1.6 Coefficient of Variation (CV)

The CV measures risk (standard deviation) per unit of expected return. It is a relative measure of dispersion, useful for comparing investments of different scales.

CV = s / ¯x

• Lower CV → better risk-return trade-off
• CV is unitless, allowing comparison across different assets
• Only meaningful when ¯x > 0 (positive mean)

Example: Investment A has mean 10% and s = 15% → CV = 1.50. Investment B has mean 20% and s = 25% → CV = 1.25. Investment B offers better risk per unit of return.

1.7 Sharpe Ratio

The Sharpe ratio measures excess return per unit of total risk. It is a reward-to-risk ratio.

Sharpe Ratio = (¯R_p − R_f) / s_p

Where: - ¯R_p = mean portfolio return - R_f = risk-free rate of return - s_p = standard deviation of portfolio returns

Key points: - Higher Sharpe ratio → better risk-adjusted performance - The numerator is the excess return (above the risk-free rate) - The denominator uses total risk (standard deviation), not just systematic risk - The Sharpe ratio can be negative; in that case, ranking by CV may be more informative

Important distinction: The CV uses standard deviation / mean (total risk per unit of total return). The Sharpe ratio uses (excess return) / standard deviation (excess return per unit of total risk). The Sharpe ratio is more commonly used in portfolio evaluation.

1.8 Population vs. Sample

The goal of statistical inference is to estimate population parameters using sample statistics.

2. Comparison Table: CV vs Sharpe Ratio

3. Weekly Quiz — 10 Practice Questions

Question 1

A dataset contains the following annual returns: 8%, 12%, −3%, 15%, 6%. What is the arithmetic mean return? - A) 7.2% - B) 7.6% - C) 8.4%

Question 2

An investment grew from $100 to $150 over 3 years. The geometric mean annual return (CAGR) is closest to: - A) 14.47% - B) 16.67% - C) 16.96%

Question 3

The 25th percentile (Q1) of the dataset {2, 5, 8, 12, 15, 18, 21, 25} is closest to: - A) 5.75 - B) 6.50 - C) 8.00

Question 4

A portfolio has a mean return of 12% and a standard deviation of 20%. According to Chebyshev's inequality, at least what percentage of returns lie within 2.5 standard deviations of the mean? - A) 75% - B) 84% - C) 96%

Question 5

The coefficient of variation (CV) for an investment with mean return 8% and standard deviation 12% is: - A) 0.67 - B) 1.50 - C) 1.33

Question 6

Investment X has a Sharpe ratio of 0.45 and Investment Y has a Sharpe ratio of 0.60. The risk-free rate is 3%. Which statement is correct? - A) Investment X has better risk-adjusted performance - B) Investment Y has better risk-adjusted performance - C) The Sharpe ratio cannot be compared without knowing the portfolio standard deviations

Question 7

A distribution of monthly returns has excess kurtosis of +2.5. This distribution is: - A) Leptokurtic with heavier tails than the normal distribution - B) Platykurtic with lighter tails than the normal distribution - C) Mesokurtic with the same tail thickness as the normal distribution

Question 8

A distribution has a mean of 50, median of 55, and mode of 60. This distribution is: - A) Positively skewed - B) Negatively skewed - C) Symmetric

Question 9

When computing the sample variance, the denominator uses (n − 1) rather than n because: - A) The sample mean is always larger than the population mean - B) Using (n − 1) provides an unbiased estimator of the population variance - C) Sample sizes are always smaller than population sizes

Question 10

Which measure of central tendency is most appropriate for highly skewed data? - A) Arithmetic mean - B) Median - C) Geometric mean

4. Formula Summary Sheet

End of Lesson 114 — Descriptive Statistics Comprehensive Review

Next: Lesson 115 — Probability Basics: Random Variables & Probability Distributions