2.7 Descriptive Statistics

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

Average portfolio return: Statistical Tool - Mean

Typical investment outcome: Statistical Tool - Median

Most common return: Statistical Tool - Mode

Investment risk: Statistical Tool - Standard deviation

Return variability: Statistical Tool - 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 = sum of x_i / n

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:

1. Arrange values from lowest to highest

2. Find the middle value

3. 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

Symmetrical data: Best Measure - Mean. Why: Most precise

Skewed data/outliers: Best Measure - Median. Why: Not distorted by extremes

Categorical data: Best Measure - Mode. Why: Only option

Effect of Outliers

Consider returns: 5%, 7%, 8%, 9%, 50%

Mean: 15.8%. Affected by 50%? Yes - pulled up significantly

Median: 8%. Affected by 50%? No - still the middle value

Mode: None. Affected by 50%? 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

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.

Variance = sum of (x_i - mean)^2 / n

Steps:

1. Calculate the mean

2. Find each value's deviation from the mean

3. Square each deviation

4. 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.

Standard deviation = square root of [sum of (x_i - mean)^2 / n]

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:

Mean plus/minus 1 standard deviation: Approximately 68%

Mean plus/minus 2 standard deviations: Approximately 95%

Mean plus/minus 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% plus/minus 8%)

- 95% of returns fall between -6% and 26% (10% plus/minus 16%)

- 99.7% of returns fall between -14% and 34% (10% plus/minus 24%)

Application to Investment Analysis

Standard Deviation as Risk Measure

Fund A: Average Return 10%; Standard Deviation 5%; Risk Level: Lower risk

Fund B: Average Return 10%; Standard Deviation 15%; Risk Level: 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 = (Portfolio return - Risk-free rate) / Portfolio standard deviation

Where Portfolio return = R_p, Risk-free rate = R_f, Portfolio standard deviation = sigma_p

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:

1. Calculate mean, median, and mode from data

2. Understand when to use each measure

3. Recognize that standard deviation measures risk/volatility

4. Apply the 68-95-99.7 rule for normal distributions

5. Compare investments using standard deviation

Common formats include calculating mean or median from data or interpreting standard deviation.

Key Takeaways

- Mean: 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: square root of [sum of (x_i - mean)^2 / n] - 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