## What Is Variance?

Variance (σ^{2}) in statistics is a measurement of the spread between numbers in a data set. That is, it measures how far each number in the set is from the mean and therefore from every other number in the set.

### Key Takeaways

- In investing, variance is used to compare the relative performance of each asset in a portfolio.
- Because the results can be difficult to analyze, standard deviation is often used instead of variance.
- In either case, the goal for the investor is to improve asset allocation.

In investing, the variance of the returns among assets in a portfolio is analyzed as a means of achieving the best asset allocation. The variance equation, in financial terms, is a formula for comparing the performance of the elements of a portfolio against each other and against the mean.

## Understanding Variance

Variance is calculated by taking the differences between each number in the data set and the mean, then squaring the differences to make them positive, and finally dividing the sum of the squares by the number of values in the data set.

### The Formula for Variance Is

$\begin{aligned} &\text{variance } \sigma^2 =\frac{ \sum_{i=1}^n{\left(x_i - \bar{x}\right)^2} }{n} \\ &\textbf{where:}\\ &x_i=\text{the } i^{th} \text{ data point}\\ &\bar{x}=\text{the mean of all data points}\\ &n=\text{the number of data points}\\ \end{aligned}$

Variance is one of the key parameters in asset allocation, along with correlation. Calculating the variance of asset returns helps investors to develop better portfolios by optimizing the return-volatility trade-off in each of their investments.

The square root of the variance is the standard deviation (σ).

## How to Use Variance

Variance measures variability from the average or mean. To investors, variability is volatility, and volatility is a measure of risk. Therefore, the variance statistic can help determine the risk an investor assumes when purchasing a specific security.

A large variance indicates that numbers in the set are far from the mean and from each other, while a small variance indicates the opposite.

Variance can be negative. A variance value of zero indicates that all values within a set of numbers are identical.

All variances that are not zero will be positive numbers.

## Advantages and Disadvantages of Variance

Statisticians use variance to see how individual numbers relate to each other within a data set, rather than using broader mathematical techniques such as arranging numbers into quartiles.

One drawback to variance is that it gives added weight to outliers, the numbers that are far from the mean. Squaring these numbers can skew the data.

Variance can be negative. A zero value means that all of the values within a data set are identical.

The advantage of variance is that it treats all deviations from the mean the same regardless of their direction. The squared deviations cannot sum to zero and give the appearance of no variability at all in the data.

The drawback of variance is that it is not easily interpreted. Users of variance often employ it primarily in order to take the square root of its value, which indicates the standard deviation of the data set.

### Variance in Investing

Variance is a key parameter in asset allocation. Used along with correlation, determining the variance of assets can help an investor develop a portfolio that optimizes the return-volatility trade-off.

That said, risk or volatility is often expressed as a standard deviation rather than variance because the former is more easily interpreted.

## Example of Variance

Let's consider a hypothetical investing example: Returns for a stock are 10% in Year 1, 20% in Year 2, and -15% in Year 3. The average of these three returns is 5%. The differences between each return and the average are 5%, 15%, and -20% for each consecutive year.

Squaring these deviations yields 25%, 225%, and 400%, respectively. Summing these squared deviations gives 650%. Dividing the sum of 650% by the number of returns in the data set (3 in this case) yields the variance of 216.67%. Taking the square root of the variance yields the standard deviation of 14.72% for the returns.

Notably, when calculating a sample variance to estimate a population variance, the denominator of the variance equation becomes N - 1 so that the estimation is unbiased and does not underestimate the population variance.