Variance vs. Standard Deviation: Understanding the Difference

Variance and standard deviation are two of the most commonly used measures of dispersion. At first glance, they seem almost identical, and as we'll see in more detail later, they are indeed mathematically related, but they serve slightly different purposes.

But before exploring the "variance vs. standard deviation" subject further, you may want to familiarize yourself with the variance 🇺🇸 and standard deviation 🇺🇸 calculators.

What is Variance?

Variance is a statistical measure that quantifies the dispersion of values relative to the mean. The higher the variance, the further the values are from the mean. In descriptive statistics, variance can be defined as the mean of the squares of the deviations from the mean.

where:

• $\sigma^2$ — Variance;
• $\mu$ — Mean; and
• $x_i$ — i^th data point out of N total data points.

Sometimes, only a sample of the population is measured for practical reasons (e.g., scientific experiments). However, the usual variance formula seen above underestimates the population variance. Thus, when sample data is used, we replace $N$ with $n-1$ in the variance formula to avoid underestimating the variance of the population. This adjustment is known as Bessel's correction.

The formula for sample variance then becomes:

where:

• $s^2$ — Estimate of the variance;
• $\bar{x}$ — Sample mean; and
• $x_i$ — i^th data point out of N total data points.

What is standard deviation?

Like the variance, the standard deviation measures the variability of a data set. In short, it also describes the spread of the data around the mean. However, the problem with variance is that it's expressed in squared units. An easy solution is to take the square root of the variance, corresponding to the standard deviation. This gives us the following formula for standard deviation ($\sigma$):

where:

• $\sigma$ — Standard deviation;
• $\mu$ — Mean; and
• $x_i$ — i^th data point out of N total data points.

Again, if you know the entire population, divide by $N$. If you're using a sample, divide by $n - 1$.

Variance is the average 🇺🇸 of the squares of the deviations from the mean. It is a more mathematical measure often used in statistical theory and modeling. On the other hand, standard deviation, in the same units as your data, measures how far the values are from the mean. It is easier to interpret and is the version most people use in reports and real-world contexts.

Here's a quick comparison table to better understand variance vs. standard deviation:

A note on sample variance vs. standard deviation

When working with a sample, the variance and standard deviation formulae use $n−1$ instead of $N$ as denominators to ensure that the sample variance and standard deviation are unbiased estimates of the population values. Thus, sample variance measures the spread in a sample, and sample standard deviation quantifies the square root of the sample variance.

Another way to measure dispersion is the mean absolute deviation 🇺🇸 (MAD). Unlike variance and standard deviation, which square the differences from the mean, MAD simplifies things. It takes the average distance between each data point and the mean (or another central value). In other words, MAD expresses, on average, how far your numbers are from the "center" of the data set. If you are wondering how to find the mean absolute deviation, here's its formula:

where:

• $n$ — Number of values in the dataset;
• $x_i$ — i^th number of the set; and
• $m$ — Central point (usually the mean, but it can also be the median or mode).

In short:

• Variance is a mathematical measure that uses squared units and is very sensitive to outliers.
• Standard deviation is directly based on variance (its square root), meaning it is expressed in the same units as your data and is much easier to interpret.
• Finally, the mean absolute deviation is the most intuitive of the three, since it simply calculates the average of the distances from the "center" of the data set. Still, it is less commonly used in advanced statistical work.

Variance and standard deviation are very closely related. While variance gives you a mathematical measure of dispersion, standard deviation translates that dispersion into the same units as your data, making it easier to interpret. Standard deviation is generally preferred in practice because it is more straightforward, but variance remains very important in theory and statistical formulas.