# Dispersion in Statistics: Understanding How It's Used

## What Is Dispersion?

The field of statistics is used across every sector and industry to help people better understand, and predict, potential outcomes. In finance, investors often turn to statistics to gain a sense of how returns on certain assets, or groups of assets, could be distributed. This range of possible investment returns is called dispersion. In other words, dispersion refers to the range of potential outcomes of investments based on historical volatility or returns.

There are two important ways to measure dispersion—alpha and beta—which calculate risk-adjusted returns and returns relative to a benchmark, respectively. By considering the dispersion of possible investment returns and values such as alpha and beta, investors can gain a sense of the risk inherent in a particular security or investment portfolio.

### Key Takeaways

• Dispersion refers to the range of potential outcomes of investments based on historical volatility or returns.
• Dispersion can be measured using alpha and beta, which calculate risk-adjusted returns and returns relative to a benchmark index, respectively.
• Generally speaking, the higher the dispersion, the riskier an investment is, and vice versa.

## Understanding Dispersion

Dispersion is often interpreted as a measure of the degree of uncertainty, and thus risk, associated with a particular security or investment portfolio.

Investors have thousands of potential securities to invest in and many factors to consider in choosing where to invest. One factor high on their list of considerations is the risk profile of the investment. Dispersion is one of many statistical measures to give perspective.

Most funds will address their risk profile in their fact sheets or prospectuses, which can be readily found on the internet. Information on individual stocks, meanwhile, can be found via Morningstar and similar stock rating companies.

When it comes to statistics in finance, investors often turn to metrics such as correlation when discussing diversification and the variation of portfolios over time. However, according to S&P Dow Jones Indices, the measure known as asset dispersion has strong qualifications as a complementary tool. Understandably, since the dispersion of possible returns on an asset provides insight about the volatility and risk associated with holding that asset. The more variable the return on an asset, the more risky or volatile it is.

For example, an asset whose historical return in any given year ranges from +10% to -10% can be considered more volatile than an asset whose historical return ranges from +3% to -3% because its returns are more widely dispersed.

## Measuring Dispersion

### Beta

The primary risk measurement statistic, beta, measures the dispersion of a security's return relative to a particular benchmark or market index, most frequently the U.S. S&P 500 index. A beta measure of 1.0 indicates the investment moves in unison with the benchmark.

A beta greater than 1.0 indicates the security is likely to experience moves greater than the market as a whole—a stock with a beta of 1.3 could be expected to experience moves that are 1.3x the market, meaning if the market is up 10%, the beta stock of 1.3 climbs 13%. The flip side is that, if the market goes down, that security will likely go down more than the market, although there are no guarantees of the magnitude of the moves.

A beta of less than 1.0 signifies a less dispersed return relative to the overall market. For example, a security with a beta of 0.87 will likely trail the overall market—if the market is up 10%, then the investment with the lower beta would be expected to rise only 8.7%.

### Alpha

Alpha is a statistic that measures a portfolio's risk-adjusted returns—that is, how much, more or less, the investment returned relative to the index or beta.

A return higher than the beta indicates a positive alpha, usually attributed to the success of the portfolio manager or model. A negative alpha, on the other hand, indicates the lack of success of the portfolio manager in beating the beta or, more broadly, the market.

## What Is Descriptive Stats?

Descriptive statistics is a means of using summaries of a data sample to describe features of a larger data set. For example, a population census may include descriptive statistics regarding the ratio of men and women in a specific city.

## What Is Covariance?

Covariance is a statistical measure of the directional relationship between two asset returns. Finding that two stocks that have a high or low covariance might not be a useful metric on its own. Some investors may choose to supplement their analysis by looking at the dispersion of returns, or correlation before deciding to invest.

## What Is Beta?

In finance, beta is a measure of volatility of a security or portfolio compared to the market as a whole. Tracking beta over time can provide investors with a useful risk profile for the asset compared to a major index.

## The Bottom Line

Dispersion refers to a statistical measure of the range of potential outcomes for an investment based on its historical volatility or returns. Two important ways to represent dispersion are alpha, which calculates risk-adjusted returns, and beta, which describes returns relative to a benchmark.
Analyzing the dispersion of possible returns can help you understand the level of risk represented by a particular investment, although it's important to keep in mind that a security's future returns may diverge from its past performance.

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1. S&P Dow Jones Indices. "Dispersion: Measuring Market Opportunity."