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# Variance

## What Is Variance?

The term variance refers to a statistical measurement of the spread between numbers in a data set. More specifically, variance measures how far each number in the set is from the mean and thus from every other number in the set. Variance is often depicted by this symbol: σ2. It is used by both analysts and traders to determine volatility and market security.

The square root of the variance is the standard deviation (σ), which helps determine the consistency of an investment’s returns over a period of time.

### Key Takeaways

• Variance is a measurement of the spread between numbers in a data set.
• Investors use variance to see how much risk an investment carries and whether it will be profitable.
• Variance is also used to compare the relative performance of each asset in a portfolio to achieve the best asset allocation.
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## Understanding Variance

In statistics, variance measures variability from the average or mean. It 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.

Variance is calculated by using the following formula:

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

A large variance indicates that numbers in the set are far from the mean and far from each other. A small variance, on the other hand, indicates the opposite. A variance value of zero, though, indicates that all values within a set of numbers are identical. Every variance that isn’t zero is a positive number. A variance cannot be negative. That’s because it’s mathematically impossible since you can’t have a negative value resulting from a square.

Variance is an important metric in the investment world. Variability is volatility, and volatility is a measure of risk. It helps assess the risk that investors assume when they buy a specific asset and helps them determine whether the investment will be profitable. But how is this done? Investors can analyze the variance of the returns among assets in a portfolio to achieve the best asset allocation. In financial terms, the variance equation is a formula for comparing the performance of the elements of a portfolio against each other and against the mean.

## Special Considerations

You can also use the formula above to calculate the variance in areas other than investments and trading, with some slight alterations. For instance, 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.