Study guide
2.7 Descriptive Statistics
Series 65 Uniform Investment Adviser Law
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1 Introduction
About the Series 65 How to Use This Guide
2 Chapter 1: Economic Factors & Business Cycles
1.1 Business Cycles 1.2 Monetary Policy 1.3 Fiscal Policy 1.4 Inflation & Deflation 1.5 Interest Rates & Yield Curves 1.6 Global Economic Factors
3 Chapter 2: Financial Reporting & Analytical Methods
2.1 Financial Statements Overview 2.2 Accounting Fundamentals 2.3 SEC Filings & Annual Reports 2.4 Financial Ratios 2.5 Time Value of Money 2.6 IRR and NPV 2.7 Descriptive Statistics
4 Chapter 3: Types of Risk
3.1 Systematic (Market) Risk 3.2 Unsystematic (Company-Specific) Risk 3.3 Interest Rate & Reinvestment Risk 3.4 Inflation & Currency Risk 3.5 Liquidity Risk & Opportunity Cost 3.6 Risk Measurement & Portfolio Theory
5 Chapter 4: Cash Equivalents & Fixed Income Securities
4.1 Cash & Cash Equivalents 4.2 Treasury Securities 4.3 Agency Securities 4.4 Corporate Bonds 4.5 Municipal Bonds 4.6 Bond Pricing & Yields
6 Chapter 5: Equity Securities
5.1 Common Stock 5.2 Preferred Stock 5.3 Shareholder Rights & Corporate Actions 5.4 Equity Valuation Methods 5.5 ADRs, REITs & Other Equity Securities
7 Chapter 6: Pooled Investments
6.1 Mutual Funds 6.2 Mutual Fund Costs & Fees 6.3 Exchange-Traded Funds (ETFs) 6.4 Closed-End Funds & UITs 6.5 Hedge Funds & Private Equity
8 Chapter 7: Derivatives, Alternative Investments & Insurance Products
7.1 Options Basics 7.2 Options Strategies 7.3 Alternative Investments 7.4 Annuities 7.5 Life Insurance Products
9 Chapter 8: Client Types & Profiles
8.1 Individual Clients 8.2 Business Entity Clients 8.3 Trusts & Estates 8.4 Accredited & Qualified Investors 8.5 Suitability & Know Your Customer
10 Chapter 9: Portfolio Theory & Capital Markets
9.1 Modern Portfolio Theory 9.2 Capital Asset Pricing Model (CAPM) 9.3 Efficient Market Hypothesis 9.4 Risk & Performance Measures 9.5 Asset Allocation
11 Chapter 10: Portfolio Management Strategies
10.1 Investment Styles 10.2 Portfolio Management Techniques 10.3 Fixed Income Strategies 10.4 Return Calculations
12 Chapter 11: Tax Considerations & Retirement Planning
11.1 Capital Gains & Investment Taxes 11.2 Retirement Plans Overview 11.3 Individual Retirement Accounts (IRAs) 11.4 Education Savings Plans 11.5 Distributions & RMDs
13 Chapter 12: ERISA, Special Accounts & Performance Measures
12.1 ERISA Requirements 12.2 Special Account Types 12.3 Performance Measures & Reporting 12.4 QDROs & Special Situations
14 Chapter 13: Regulation of Investment Advisers
13.1 Investment Advisers Act of 1940 13.2 Registration Requirements 13.3 Recordkeeping Requirements 13.4 Custody Requirements 13.5 Exempt Reporting Advisers
15 Chapter 14: Regulation of Investment Adviser Representatives
14.1 IAR Definition & Requirements 14.2 Registration & Post-Registration 14.3 IARs of Federal Covered Advisers
16 Chapter 15: Regulation of Broker-Dealers & Agents
15.1 Broker-Dealer Definition 15.2 Agent Registration 15.3 Regulation Best Interest
17 Chapter 16: Securities Registration & Exemptions
16.1 Uniform Securities Act 16.2 Securities Registration Methods 16.3 Exempt Securities 16.4 Exempt Transactions
18 Chapter 17: Communications & Client Relationships
17.1 Advertising Rules 17.2 Social Media & Electronic Communications 17.3 Client Agreements & Disclosure 17.4 Proxy Voting & Other Obligations
19 Chapter 18: Ethics & Fiduciary Obligations
18.1 Fiduciary Duties 18.2 Prohibited Practices 18.3 Senior & Vulnerable Adult Protection 18.4 AML & Compliance Programs
2.7 Descriptive Statistics
Key Takeaways Explain This Topic AI
Mean, median, and mode describe central tendency.
Standard deviation measures dispersion of returns.
Variance is standard deviation squared.
Skewness and kurtosis describe distribution shape.
Correlation shows relationship between two variables.
Last updated: October 2026
Descriptive statistics help investment advisers summarize and interpret investment data. Understanding measures of central tendency and dispersion is essential for portfolio analysis, risk assessment, and client communication.
Why Statistics Matter for Investment Advisers
- Application Statistical Tool
- Average portfolio return Mean
- Typical investment outcome Median
- Most common return Mode
- Investment risk Standard deviation
- Return variability Range
- Measures of Central Tendency
Central tendency measures describe the "middle" or "typical" value of a data set.
Mean (Average)
The mean is the sum of all values divided by the number of values. Mean = ∑ x i n \text{Mean} = \frac{\sum x_i}{n} Mean = n ∑ x i
- Example : Portfolio returns of 5%, 8%, 12%, 3%, 7%
- Mean = (5 + 8 + 12 + 3 + 7) / 5 = 35 / 5 = 7%
Characteristics:
- Most commonly used average
- Affected by extreme values (outliers)
- Best when data is symmetrically distributed
- Median
The median is the middle value when data is arranged in order.
Steps :
- Arrange values from lowest to highest
- Find the middle value
- If even number of values, average the two middle values
- Example : Returns of 3%, 5%, 7%, 8%, 12%
- Middle value = 7% (median)
- With an even number : Returns of 3%, 5%, 7%, 8%
- Median = (5% + 7%) / 2 = 6%
Characteristics:
- Not affected by extreme values
- Better measure when data has outliers
- Often used for income data (where extremes distort mean)
- Mode
The mode is the value that appears most frequently.
- Example : Returns of 5%, 7%, 7%, 8%, 12%
- Mode = 7% (appears twice)
Characteristics:
- Data can have no mode (all values different)
- Data can have multiple modes (bimodal, multimodal)
- Only measure usable for categorical data
- Comparing Mean, Median, and Mode
- Scenario Best Measure Why
- Symmetrical data Mean Most precise
- Skewed data/outliers Median Not distorted by extremes
- Categorical data Mode Only option
- Effect of Outliers
- Consider returns: 5%, 7%, 8%, 9%, 50%
Measure Value Affected by 50%?
- Mean 15.8% Yes—pulled up significantly
- Median 8% No—still the middle value
- Mode None N/A
The mean of 15.8% misrepresents typical returns because one outlier (50%) distorts it.
Measures of Dispersion
Dispersion measures describe how spread out the data is.
Range
The range is the difference between the highest and lowest values. Range = Maximum Value − Minimum Value \text{Range} = \text{Maximum Value} - \text{Minimum Value} Range = Maximum Value − Minimum Value
- Example : Returns of 3%, 5%, 7%, 8%, 12%
- Range = 12% - 3% = 9 percentage points
Limitations : Only uses two data points; ignores distribution in between.
Variance
Variance measures the average squared deviation from the mean. σ 2 = ∑ ( x i − x ˉ ) 2 n \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} σ 2 = n ∑ ( x i − x ˉ ) 2
Steps :
- Calculate the mean ( x ˉ \bar{x} x ˉ )
- Find each value's deviation from the mean
- Square each deviation
- Average the squared deviations
Variance is expressed in squared units , making it difficult to interpret directly.
Standard Deviation
Standard deviation is the square root of variance—the average distance from the mean. σ = ∑ ( x i − x ˉ ) 2 n \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} σ = n ∑ ( x i − x ˉ ) 2
Why It Matters:
- Expressed in the same units as the original data
- Most common measure of investment risk
- Larger standard deviation = more volatility = more risk
Example : If a fund has a 10% average return with 15% standard deviation:
- Returns typically range from -5% to 25% (one standard deviation)
- Higher standard deviation = wider range of possible outcomes
- The Normal Distribution and Standard Deviation
When data is normally distributed (bell-shaped), standard deviation has special meaning:
- Range Percentage of Data
- Mean ± 1 standard deviation Approximately 68%
- Mean ± 2 standard deviations Approximately 95%
- Mean ± 3 standard deviations Approximately 99.7%
This is called the 68-95-99.7 Rule (or Empirical Rule).
Example : A stock has mean return of 10% and standard deviation of 8%:
- 68% of returns fall between 2% and 18% (10% ± 8%)
- 95% of returns fall between -6% and 26% (10% ± 16%)
- 99.7% of returns fall between -14% and 34% (10% ± 24%)
- Application to Investment Analysis
- Standard Deviation as Risk Measure
- Fund Average Return Standard Deviation Risk Level
- Fund A 10% 5% Lower risk
- Fund B 10% 15% Higher risk
Both funds have the same average return, but Fund B is riskier—its returns vary more widely.
Risk-Adjusted Performance
Standard deviation enables risk-adjusted comparisons: Sharpe Ratio = R p − R f σ p \text{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p} Sharpe Ratio = σ p R p − R f
Where R p R_p R p = Portfolio return, R f R_f R f = Risk-free rate, σ p \sigma_p σ p = Portfolio standard deviation
- Higher Sharpe ratio = better risk-adjusted performance
- In Practice: How Investment Advisers Apply This
Analyzing investments:
- Use standard deviation to compare fund volatility
- Consider median returns when outliers are present
- Explain risk in terms of standard deviation ranges
Client communication:
- "This fund averages 8% with returns typically between 0% and 16%"
- Use median when discussing income or home prices (skewed data)
- Explain that higher returns often come with higher standard deviation
- On the Exam
The Series 65 exam tests your ability to:
- Calculate mean, median, and mode from data
- Understand when to use each measure
- Recognize that standard deviation measures risk/volatility
- Apply the 68-95-99.7 rule for normal distributions
- Compare investments using standard deviation
Common formats include calculating mean or median from data or interpreting standard deviation.
- Key Takeaways
- Mean : ∑ x i n \frac{\sum x_i}{n} n ∑ x i — sum of values / number of values (affected by outliers)
- Median = Middle value when data is ordered (not affected by outliers)
- Mode = Most frequently occurring value
- Range = Maximum - Minimum (simple but limited)
Standard Deviation : σ = ∑ ( x i − x ˉ ) 2 n \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} σ = n ∑ ( x i − x ˉ ) 2 — measures risk
- Use median when data has outliers or is skewed
- In a normal distribution: 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD
- Higher standard deviation = higher risk/volatility