Descriptive Statistics

Quick Answer

Descriptive statistics summarize investment data. Mean, median, and mode describe the center of a distribution; range and standard deviation describe its spread. Standard deviation captures total risk, beta isolates systematic risk, alpha measures manager skill, correlation describes how two securities move together, R-squared measures how much of that movement a benchmark explains, and the Sharpe ratio measures risk-adjusted return.

Each measure answers a different exam question: what is typical, how volatile is it, and did the manager add value. Knowing which measure answers which question is tested more often than the arithmetic itself.

Central tendency measures establish the "center" of a distribution so you have a baseline to measure deviation from.

Mean (Average): Definition: Sum of values divided by count. How to calculate: Add all returns, divide by number of periods. Best use: General performance summary.

Median: Definition: Middle value when sorted. How to calculate: Arrange in order, find middle number. Best use: When outliers exist.

Mode: Definition: Most frequently occurring value. How to calculate: Find the value that appears most often. Best use: Identifying common outcomes.

Which Measure Should You Use?

Mean: Best used when data is normally distributed. Limitation: Skewed by outliers (extreme values).

Median: Best used when outliers exist (skewed distribution). Limitation: Ignores the actual values (only position matters).

Mode: Best used when identifying most common outcome. Limitation: May not exist; not useful for continuous data.

When outliers are present, the median is more representative than the mean. If most portfolio returns cluster around 8 to 10 percent but one year had a 50 percent gain, the median better represents the typical year.

Which way did the outliers pull it? The gap between mean and median tells you which way the outliers sit:

Mean greater than Median: A few unusually high values pulled the mean up. This is a right-skewed (positive skew) distribution.

Mean less than Median: A few unusually low values pulled the mean down. This is a left-skewed (negative skew) distribution.

Exam Tip: Gotchas

A portfolio with a median return of 7 percent but a mean return of 4 percent is left-skewed: a few exceptionally poor returns dragged the average below the typical result.

How Do You Calculate These Measures in an Example?

Given annual returns of: 6 percent, 4 percent, 11 percent, 10 percent, 4 percent.

Mean: (6 plus 4 plus 11 plus 10 plus 4) divided by 5 equals 7 percent.

Median: Sorted: 4, 4, 6, 10, 11. Middle value equals 6 percent.

Mode: 4 percent appears twice, so mode equals 4 percent.

A data set can have no mode (no repeating values), one mode, or multiple modes (two or more values tie for most frequent).

What Is the Difference Between the Arithmetic Mean and the Geometric Mean?

The mean in the table above is the arithmetic mean: the simple average. A second average, the geometric mean, fits returns that compound from one period to the next.

Arithmetic mean: How it works: Add the returns, divide by the number of periods. What it tells you: The simple average of the periods.

Geometric mean: How it works: Multiply each period's growth factor (1 plus return), take the nth root, subtract 1. What it tells you: The compounded growth rate you actually earned per period.

The two means are equal only when every period's return is identical.

When returns vary, the geometric mean is lower than the arithmetic mean. Volatility drags down compounded growth.

The signs of the returns do not change this rule. Mostly positive returns that still vary give a geometric mean below the arithmetic mean.

Example: A fund gains 10 percent in one year and loses 10 percent the next. The arithmetic mean is 0 percent. But 1 dollar grows to 1.10 dollars, then falls to 0.99 dollars, so the geometric mean is about minus 0.5 percent per year. Three years of plus 8 percent give 8 percent under both means, because nothing varies.

For the geometric mean formula, annualized return, and CAGR (compound annual growth rate), see the Returns section of the Portfolio Performance Measures unit.

Exam Tip: Gotchas

If a question asks when the geometric mean equals the arithmetic mean, the answer is identical returns in every period. Having positive returns, alternating returns, or more up periods than down periods does not make them equal.

What Does Range Measure?

Range equals highest value minus lowest value.

Range: Definition: Spread between highest and lowest. How to calculate: Highest return minus lowest return. Best use: Quick volatility estimate.

Example: Using the same returns (6 percent, 4 percent, 11 percent, 10 percent, 4 percent): Range equals 11 percent minus 4 percent equals 7 percent.

Range is simple but limited; it only considers two data points.

What Does Standard Deviation Measure?

Standard deviation quantifies total risk: both systematic and unsystematic.

Higher standard deviation equals more volatile equals higher risk.

Lower standard deviation equals more stable equals lower risk.

How Does the Normal Distribution (Bell Curve) Apply?

68 percent of returns fall within plus or minus 1 standard deviation from the mean.

95 percent of returns fall within plus or minus 2 standard deviations from the mean.

99.7 percent of returns fall within plus or minus 3 standard deviations from the mean.

Example: A security with an 8.7 percent expected return and 14.6 percent standard deviation:

68 percent chance: Returns fall between minus 5.9 percent and plus 23.3 percent (8.7 plus or minus 14.6).

95 percent chance: Returns fall between minus 20.5 percent and plus 37.9 percent (8.7 plus or minus 29.2).

How Do You Compare Investments by Volatility?

Company A: Returns over 4 years 12 percent, 4 percent, 8 percent, 6 percent. Mean 7.5 percent. Deviation from mean minus 3.5 percent to plus 4.5 percent. Volatility very low.

Company B: Returns 7 percent, 8 percent, 9 percent, 6 percent. Mean 7.5 percent. Deviation minus 1.5 percent to plus 1.5 percent. Volatility lowest.

Company C: Returns 10 percent, 12 percent, minus 2 percent, 10 percent. Mean 7.5 percent. Deviation minus 9.5 percent to plus 4.5 percent. Volatility moderate.

Company D: Returns 15 percent, 20 percent, minus 8 percent, 3 percent. Mean 7.5 percent. Deviation minus 15.5 percent to plus 12.5 percent. Volatility highest.

An investment that moves up 5 percent, then 6 percent, then 4 percent has low standard deviation: consistent returns. One that moves up 30 percent, down 20 percent, then up 15 percent has high standard deviation: wide swings. Both can share the same average return; one is a smooth ride, the other a roller coaster.

Exam Tip: Gotchas

Standard deviation measures total risk (systematic plus unsystematic). Beta measures only systematic risk. If asked which measure captures total risk, the answer is standard deviation.

What Does Correlation Tell You?

Correlation measures the strength and direction of the linear relationship between two securities' returns. The correlation coefficient (r) ranges from minus 1.0 to plus 1.0.

How Do You Read the Correlation Coefficient?

Plus 1.0: Perfect positive correlation. Two securities move together in perfect lockstep.

0: No correlation. No linear relationship (independence is not implied).

Minus 1.0: Perfect negative correlation. Two securities move in exactly opposite directions.

Between: Partial correlation. Strength increases as coefficient approaches plus or minus 1.0.

At plus 1.0, if Security A rises 5 percent, Security B rises by a proportional amount. At 0, there is no linear relationship between the two, though they can still be related in a non-linear way.

Which Pair Has the Strongest Relationship?

Given these correlation coefficients, which pair moves together most predictably, regardless of direction?

Assets A and B: correlation plus 0.90.

Assets C and D: correlation plus 0.47.

Assets E and F: correlation 0.

Assets G and H: correlation minus 0.88.

Answer: A and B (plus 0.90) has the strongest positive relationship. G and H (minus 0.88) has a nearly as strong negative relationship.

Why Does Correlation Matter for Portfolio Construction?

Securities with low or negative correlation move more independently of each other. Combining assets with low correlation can reduce overall portfolio volatility. (Portfolio construction strategies are covered in the Portfolio Management Strategies unit.)

What Does R-Squared Add?

R-squared equals correlation coefficient squared.

Represents the percentage of a portfolio's movement explained by the benchmark.

R-squared of 0.90 means 90 percent of the portfolio's returns are explained by the benchmark.

A higher R-squared makes beta more meaningful as a risk measure.

Exam Tip: Gotchas

Correlation of plus 1.0 provides NO diversification benefit. Maximum diversification benefit comes from combining assets with negative correlation. Zero correlation still provides some diversification benefit.

What Does Beta Measure?

While standard deviation measures total risk, beta isolates systematic (market) risk only. Beta measures a security's sensitivity to movements in the overall market (the S and P 500 has a beta of 1.0).

Beta 0: No market correlation. Example: 91-day T-bill.

Beta less than 1: Moves less than market. Example: Utilities, defensive stocks.

Beta equals 1: Moves with market. Example: Index funds.

Beta greater than 1: Moves more than market. Example: Tech stocks, small caps.

Beta 2.0: Moves about twice as much as market. Example: Highly leveraged funds.

Negative beta: Moves opposite to market. Example: Rare; some gold stocks.

Calculations:

If beta equals 1.2 and the market rises 10 percent, the stock rises 12 percent (10 percent times 1.2).

If beta equals 0.85 and the market falls 10 percent, the stock falls 8.5 percent (10 percent times 0.85).

A beta of 2.0 means the security tends to move about twice as much as the market, common among highly leveraged funds.

If the market is an ocean and stocks are boats, beta tells you how much a boat rocks when waves hit. A beta of 1.0 rocks exactly as much as the water moves; 1.5 rocks 50 percent more than the waves; 0.5 rocks only half as much. Beta does not measure how good the boat is, only how it responds to the market.

Exam Tip: Gotchas

A stock with a beta of 1.0 still has unsystematic risk; it just moves with the market on average. Diversification reduces unsystematic risk but does NOT reduce beta.

What Does Alpha Measure?

Alpha measures the excess return of an investment relative to its expected return based on its risk level (CAPM).

Formula: Alpha equals Actual Return minus Expected Return.

Alpha equals R portfolio minus [R f plus Beta times (R market minus R f)].

R f equals risk-free rate; R market equals market return.

Positive alpha: the investment outperformed its risk-adjusted expectation (manager added value).

Negative alpha: the investment underperformed its risk-adjusted expectation.

Zero alpha: the investment performed exactly as expected for its risk level.

Example: A portfolio with a beta of 1.1 earned 14 percent when the expected return was 12.8 percent. Alpha equals plus 1.2 percent, meaning the manager added 1.2 percent of value above what the portfolio's risk alone would have generated.

Exam Tip: Gotchas

Alpha is not simply how much an investment returned; it is how much EXTRA it returned above what was expected given its risk (beta). A fund that returned 12 percent with an expected return of 10 percent has an alpha of plus 2 percent, even if another fund returned 15 percent at higher risk.

What Does the Sharpe Ratio Measure?

Raw returns tell only part of the story; two portfolios with identical returns can carry very different risk. The Sharpe ratio is a risk-adjusted performance measure that lets you compare investments on equal footing.

Purpose: Measures excess return per unit of total risk (standard deviation).

Formula: Sharpe Ratio equals (R p minus R f) divided by sigma p.

Where R p equals portfolio return, R f equals risk-free rate (typically the 91-day/3-month Treasury bill rate), sigma p equals standard deviation of the portfolio (total risk). The numerator is called the excess return.

How Do You Compare Two Portfolios?

Portfolio A: Return 12 percent. Risk-free rate 2 percent. Std Dev 20 percent. Sharpe Ratio (12 percent minus 2 percent) divided by 20 percent equals 0.50.

Portfolio B: Return 14 percent. Risk-free rate 2 percent. Std Dev 16 percent. Sharpe Ratio (14 percent minus 2 percent) divided by 16 percent equals 0.75.

A higher Sharpe ratio always means better risk-adjusted performance, so Portfolio B wins the comparison above.

Exam Tip: Gotchas

Sharpe uses standard deviation (total risk). If asked which ratio uses total risk, the answer is Sharpe. It is the appropriate measure when the portfolio is the investor's entire holding, because unsystematic risk has not been diversified away.

How Do the Risk Measures Compare?

Standard Deviation: Measures dispersion of returns around the mean. Risk type: Total risk. Key formula component: Variance of returns.

Beta: Measures sensitivity to market movements. Risk type: Systematic risk. Key formula component: Covariance with market.

Alpha: Measures excess return vs. expected (CAPM). Risk type: Manager skill. Key formula component: Actual minus Expected return.

Sharpe Ratio: Measures return per unit of total risk. Risk type: Total risk. Key formula component: Uses standard deviation.

R-squared: Measures percent of returns explained by benchmark. Risk type: Correlation. Key formula component: Correlation coefficient squared.

What Should You Check on Exam Day?

Standard deviation captures total risk; beta captures systematic risk only. Know which question is being asked before you pick an answer.

Mean is pulled toward outliers; median is not. A mean below the median signals left skew. The geometric mean equals the arithmetic mean only when every return is identical; variation puts it lower.

Correlation of plus 1.0 gives zero diversification benefit; negative correlation gives the most.

Alpha is excess return over the risk-adjusted expectation, not raw return.

Sharpe ratio's denominator is standard deviation (total risk), so it fits an investor's entire portfolio.

Additional Details

Published: 2026-11-02T21:59:23.883Z