What Is Variance in Statistics? Definition, Formula, and Example

Variance Definition

Investopedia / Alex Dos Diaz

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.

Key Takeaways

  • Variance is a measurement of the spread between numbers in a data set.
  • In particular, it measures the degree of dispersion of data around the sample's mean.
  • Investors use variance to see how much risk an investment carries and whether it will be profitable.
  • Variance is also used in finance to compare the relative performance of each asset in a portfolio to achieve the best asset allocation.
  • The square root of the variance is the standard deviation.
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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:

σ 2 = i = 1 n ( x i x ) 2 N where: x i = Each value in the data set x = Mean of all values in the data set N = Number of values in the data set \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=Ni=1n(xix)2where:xi=Each value in the data setx=Mean of all values in the data setN=Number of values in the data set

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.

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.

In some cases, risk or volatility may be expressed as a standard deviation rather than a variance because the former is often more easily interpreted.

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

How Do I Calculate Variance?

Follow these steps to compute variance:

  1. Calculate the mean of the data.
  2. Find each data point's difference from the mean value.
  3. Square each of these values.
  4. Add up all of the squared values.
  5. Divide this sum of squares by n – 1 (for a sample) or N (for the population).

What Is Variance Used for?

Variance is essentially the degree of spread in a data set about the mean value of that data. It shows the amount of variation that exists among the data points. Visually, the larger the variance, the "fatter" a probability distribution will be. In finance, if something like an investment has a greater variance, it may be interpreted as more risky or volatile.

Why Is Standard Deviation Often Used More Than Variance?

Standard deviation is the square root of variance. It is sometimes more useful since taking the square root removes the units from the analysis. This allows for direct comparisons between different things that may have different units or different magnitudes. For instance, to say that increasing X by one unit increases Y by two standard deviations allows you to understand the relationship between X and Y regardless of what units they are expressed in.