3.2 Statistics, Distributions, and Risk Measures

CFA Level I Study Guide Exam

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1 Chapter 1: Orientation, Official Sources, and Exam Strategy

1.1 What Level I Tests and Why It Matters 1.2 Official Source Control and 2026 Curriculum 1.3 Eligibility, Registration, Passport, and PSM 1.4 Exam Format, Fees, Windows, and Results 1.5 Topic Weights and Session Map 1.6 Three-Hundred-Hour Study Plan

2 Chapter 2: Ethical and Professional Standards

2.1 Ethics Weight, Code, and Standards Map 2.2 Professionalism and Integrity of Capital Markets 2.3 Duties to Clients and Employers 2.4 Investment Analysis, Recommendations, and Conflicts 2.5 Responsibilities as a CFA Candidate and Member 2.6 GIPS, Performance Presentation, and Ethics Case Lab

3 Chapter 3: Quantitative Methods

3.1 Rates, Returns, and Time Value of Money 3.2 Statistics, Distributions, and Risk Measures 3.3 Probability, Conditional Expectations, and Bayes 3.4 Portfolio Mathematics and Safety-First 3.5 Sampling, Estimation, and Hypothesis Testing 3.6 Regression, Simulation, and Big Data

4 Chapter 4: Economics

4.1 Firm Market Structures and Breakeven Analysis 4.2 Business Cycles and Economic Indicators 4.3 Fiscal Policy, Monetary Policy, and Central Banks 4.4 Geopolitics, Globalization, and Trade 4.5 Foreign Exchange, Capital Flows, and Arbitrage 4.6 Economics Case Lab

5 Chapter 5: Financial Statement Analysis

5.1 FSA Framework and Information Quality 5.2 Income Statements, Revenue, Expense, and EPS 5.3 Balance Sheets, Assets, Liabilities, and Equity 5.4 Cash Flow Statements and Free Cash Flow 5.5 Inventories, Long-Lived Assets, and Taxes 5.6 Reporting Quality, Ratios, DuPont, and Modeling

6 Chapter 6: Corporate Issuers

6.1 Organizational Forms and Corporate Features 6.2 Stakeholders, ESG, and Governance Conflicts 6.3 Working Capital, Liquidity, and Cash Conversion 6.4 Capital Budgeting, NPV, IRR, and Real Options 6.5 Capital Structure, WACC, and Modigliani-Miller 6.6 Business Models and Corporate Case Lab

7 Chapter 7: Equity Investments

7.1 Market Organization, Orders, and Trading 7.2 Security Market Indexes and Efficiency 7.3 Equity Securities and Industry Analysis 7.4 Company Analysis and Competitive Position 7.5 Dividend Discount Models and Valuation Inputs 7.6 Multiples, Private Company Valuation, and Equity Case Lab

8 Chapter 8: Fixed Income

8.1 Bond Features, Issuance, and Cash Flow Structures 8.2 Yield Measures, Spot Rates, and Forward Rates 8.3 Bond Valuation, Clean Price, and Accrued Interest 8.4 Duration, Convexity, and Interest Rate Risk 8.5 Credit Risk, Ratings, and Spreads 8.6 Securitization and Fixed Income Case Lab

9 Chapter 9: Derivatives

9.1 Derivative Markets, Forward Commitments, and Claims 9.2 Forwards, Futures, and Swap Basics 9.3 Option Payoffs, Moneyness, and Risk 9.4 Arbitrage, Pricing, and No-Arbitrage Intuition 9.5 Hedging, Speculation, and Risk Management 9.6 Derivatives Case Lab

10 Chapter 10: Alternative Investments

10.1 Alternatives Role, Characteristics, and Structures 10.2 Hedge Funds, Strategies, Fees, and Risks 10.3 Private Capital, Venture, Buyout, and Private Credit 10.4 Real Estate, Infrastructure, and Commodities 10.5 Liquidity, Due Diligence, and Performance Measurement 10.6 Alternatives Case Lab

11 Chapter 11: Portfolio Management

11.1 Portfolio Return, Risk, and Diversification 11.2 Portfolio Planning, Policy, and Construction 11.3 Risk Management and Investor Objectives 11.4 Behavioral Finance and Client Decision Errors 11.5 Fintech, Data, and Portfolio Applications 11.6 Portfolio Case Lab

12 Chapter 12: Integrated CFA Level I Review

12.1 Session One Integration: Ethics, Quant, Economics, FSA, and Corporate 12.2 Session Two Integration: Equity, Fixed Income, Derivatives, Alternatives, and PM 12.3 Formula Sheet and Calculator Workflow 12.4 Question Stem Reading and Answer Elimination 12.5 Mock Exam Review and Error Log 12.6 Mixed-Topic Case Lab

13 Chapter 13: Final Countdown, Results, and Next Steps

13.1 Ninety-Day, Sixty-Day, and Thirty-Day Plan 13.2 Exam Week Prometric and Passport Checklist 13.3 PSM Completion and Result Release 13.4 If You Pass: Level II Path 13.5 If You Fail: Retake and Deferral Strategy 13.6 Career Path, Charter Requirements, and Membership 13.7 Final Readiness Audit

3.2 Statistics, Distributions, and Risk Measures

Key Takeaways Explain This Topic AI

Descriptive statistics summarize central tendency, dispersion, shape, and relative position of return data.

Standard deviation measures total variability; downside measures (semideviation, shortfall, VaR) isolate adverse outcomes.

Skewness and kurtosis describe shape and matter because asset returns are typically nonnormal with fat tails.

The normal distribution lets analysts compute probabilities via z-scores, using 68-95-99.7 and the 1.65/1.96/2.58 critical values.

Last updated: June 2026

Statistics, Distributions, and Risk Measures

Statistics turns raw observations (returns, yields, spreads, valuation multiples) into decision-useful information. A disciplined first pass asks four questions: What is typical? How much does it vary? Is the distribution symmetric? How fat are the tails?

Central tendency and position

Measures of central tendency locate the center. The arithmetic mean is the sum over the count. The median is the middle sorted value (the average of the two middle values when N is even) and is robust to outliers. The mode is the most frequent value. When data are skewed by extremes, the median often describes the typical outcome better. Quantiles divide sorted data: quartiles into four parts, quintiles into five, deciles into ten, and percentiles into a hundred. The interquartile range, Q3 minus Q1, underlies box plots.

Dispersion

Range is maximum minus minimum but uses only two points. Mean absolute deviation (MAD) averages absolute deviations from the mean. Variance averages squared deviations; standard deviation is its square root and is reported in return units, making it the workhorse risk measure. Critically, sample variance divides by n - 1 (a degrees-of-freedom adjustment because the sample mean was estimated from the same data), while population variance divides by N . Exam stems explicitly state whether the data are a sample or a population, and the denominator choice changes the answer.

The coefficient of variation (CV) is standard deviation / mean . It measures risk per unit of expected return, so a lower CV is preferred among investments with positive expected returns. CV becomes misleading when the mean is zero or negative.

Shape: skewness and kurtosis

Returns are frequently asymmetric. Positive (right) skew has a long right tail and pulls the mean above the median; negative (left) skew has a long left tail (more dangerous for investors) and pulls the mean below the median. For a perfectly symmetric distribution, mean = median = mode. Kurtosis describes tail thickness; the normal distribution has kurtosis of 3, so excess kurtosis is kurtosis minus 3. A leptokurtic distribution (positive excess kurtosis) has fatter tails and a sharper peak, producing more extreme outcomes than the normal model predicts.

Most equity-return series are negatively skewed and leptokurtic, which is why pure normal probabilities understate crash risk.

The normal distribution and z-scores

The normal distribution is symmetric, fully described by its mean and variance, has skewness of zero and excess kurtosis of zero, and is the basis of mean-variance analysis. A z-score standardizes any observation: z = (x - mean)/standard deviation . Memorize these landmarks:

About 68% of observations lie within +/- 1 standard deviation, 95% within +/- 2, and 99.7% within +/- 3.

One-tailed critical z-values: 1.65 for 5% , 2.33 for 1% .

Two-tailed critical z-values: 1.96 for 5% (so 90% interval uses 1.65, 95% uses 1.96, 99% uses 2.58).

A return of 14% drawn from a normal distribution with mean 8% and standard deviation 3% has z = (14 - 8)/3 = 2.0 , placing it two standard deviations above the mean.

Concept Formula or meaning Candidate use
Arithmetic mean sum x / n Average return
Median Middle sorted value Outlier-robust center
Sample variance sum(x - xbar)^2 / (n-1) Total variability
Standard deviation sqrt(variance) Risk in return units
Coefficient of variation s / mean Risk per unit of return
Skewness Direction of long tail Asymmetry of outcomes
Excess kurtosis Kurtosis minus 3 Fat-tail / extreme-loss risk
z-score (x - mean)/s Standardized position
Downside risk measures

Standard deviation penalizes upside and downside equally, but investors fear losses. Semivariance uses only returns below the mean; target semideviation measures dispersion below a stated target. Shortfall risk is the probability that return falls below a threshold. Value at risk (VaR) estimates a minimum loss for a stated probability and horizon (for example, a 5% one-day VaR of 1 million means losses should exceed 1 million only 5% of days).

Match the measure to the decision: a pension with a required return cares about shortfall probability, while an option book with rare large losses needs skewness and kurtosis, not just standard deviation.

Worked dispersion example

Consider four annual returns: 4%, 8%, 12%, and 16%. The arithmetic mean is (4 + 8 + 12 + 16)/4 = 10% . The deviations from the mean are -6, -2, +2, and +6, whose squares sum to 36 + 4 + 4 + 36 = 80 . As a sample, divide by n - 1 = 3 to get a variance of 80/3 = 26.67 , so the sample standard deviation is sqrt(26.67) = 5.16% . As a population, divide by N = 4 to get a variance of 20 and a standard deviation of 4.47% . The coefficient of variation for the sample case is 5.16/10 = 0.52 , meaning roughly half a unit of return risk per unit of expected return.

This single example shows why reading sample versus population in the stem is not optional: the same data yield two different standard deviations.

Exam tactics

On the exam, if the mean exceeds the median, think positive skew; if the tails are fat (positive excess kurtosis), normal probabilities understate extreme events such as crashes; and between two funds with equal means, the one with the lower standard deviation has lower total volatility. When a question gives a target return and asks for the chance of underperforming, compute a z-score relative to that target and read the normal table; when it asks which of two assets is riskier per unit of return, compare coefficients of variation rather than raw standard deviations, because scale differs across the two return series.