Variance Definition

Variance (σ²) is a measurement of the spread between numbers in a data set. It measures how far each number in the set is from the mean and is calculated by taking the differences between each number in the set and the mean, squaring the differences (to make them positive) and dividing the sum of the squares by the number of values in the set.

Variance is one of the key parameters in asset allocation. Along with correlation, the variance of asset returns helps investors to develop optimal portfolios by optimizing the return-volatility trade-off in investment portfolios.

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

$\begin{aligned} &\text{variance } \sigma^2 =\frac{ \sum_{i=1}^n{\left(x_i - \bar{x}\right)^2} }{n} \\ &\textbf{where:}\\ &x_i=\text{the } i^{th} \text{ data point}\\ &\bar{x}=\text{the mean of all data points}\\ &n=\text{the number of data points}\\ \end{aligned}$​variance σ2=n∑i=1n​(xi​−x¯)2​where:xi​=the ith data pointx¯=the mean of all data pointsn=the number of data points​

Variance is used in statistics for probability distribution. Since variance measures the variability (volatility) from an average or mean and volatility is a measure of risk, the variance statistic can help determine the risk an investor might assume when purchasing a specific security.

A variance value of zero indicates that all values within a set of numbers are identical; all variances that are non-zero will be positive numbers. A large variance indicates that numbers in the set are far from the mean and each other, while a small variance indicates the opposite.

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. A drawback to variance is that it gives added weight to numbers far from the mean (outliers), since squaring these numbers can skew interpretations of the data.

The advantage of variance is that it treats all deviations from the mean the same regardless of direction; as a result, 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, and the square root of its value is usually taken to get the standard deviation of the data set in question.

Variance is one of the key parameters in asset allocation. Along with correlation, the variance of asset returns helps investors to develop optimal portfolios by optimizing the return-volatility trade-off in investment portfolios. Risk or volatility is often expressed as a standard deviation rather than variance because the former is more easily interpreted.

• Variance is a measurement of the spread between numbers in a data set.
• The variance measures how far each number in the set is from the mean.
• The square root of variance is the standard deviation.

Let's consider a hypothetical 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 population variance.