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 (average), 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 (SD or σ), which helps determine the consistency of an investment’s returns over a period of time.

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. Software like Excel can make this calculation easier.

Variance is calculated by using the following formula:

$\begin{aligned}&\sigma^2 = \frac { \sum_{i = 1} ^ { n } \big (x_i - \overline { x } \big ) ^ 2 }{ N } \\&\textbf{where:} \\&x_i = \text{Each value in the data set} \\&\overline { x } = \text{Mean of all values in the data set} \\&N = \text{Number of values in the data set} \\\end{aligned}$​σ2=N∑i=1n​(xi​−x)2​where:xi​=Each value ;in the data setx=Mean of all values in the data setN=Number of values in the data set​

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. The advantage of variance is that it treats all deviations from the mean as 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.

One drawback to variance, though, is that it gives added weight to outliers. These are the numbers far from the mean. Squaring these numbers can skew the data. Another pitfall of using variance is that it is not easily interpreted. Users often employ it primarily to take the square root of its value, which indicates the standard deviation of the data. As noted above, investors can use standard deviation to assess how consistent returns are over time.

Example of Variance in Finance

Here’s a hypothetical example to demonstrate how variance works. Let’s say returns for stock in Company ABC 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 0.25%, 2.25%, and 4.00%, respectively. If we add these squared deviations, we get a total of 6.5%. When you divide the sum of 6.5% by one less the number of returns in the data set, as this is a sample (2 = 3-1), it gives us a variance of 3.25% (0.0325). Taking the square root of the variance yields a standard deviation of 18% (√0.0325 = 0.180) for the returns.

Frequently Asked Questions