Probability and Statistics: What Is Variance?

Variance is a very useful calculation in statistics. It quantitatively tells us how spread out a dataset is with respect to its average, and is given by the formula $$\sigma^2=\sum\limits_{i=1}^{n}\frac{(X_i-\overline{X})^2}{n}$$ where $\sigma^2$ is the variance, $X_i$ is the $i$th element, $\overline{X}$ is the mean, and $n$ is the size of the dataset. (Note: variance is denoted as $\sigma^2$ because it is the square of another important statistical calculation, the standard deviation.)

Let's talk through the formula to make it feel a bit less confusing.

• If we want to know how spread out a dataset is from the mean, it's useful to know the difference between each term and the mean. This is what $X_i - \overline{X}$ accomplishes.
• The reason $(X_i - X)^2$ is the square of the aforementioned difference is we want our measure of spread to be a positive value. The square flips the sign of the differences between the mean and elements less than the mean, making them positive and unifying everything.
• The summation $\sum\limits_{i-1}^{n}$ sums all these squares of differences across the terms in the dataset.
• Finally, the $n$ in the denominator turns the sum we obtained above into an average.

So, variance is ultimately just the average of the squares of differences between the mean of a dataset and each element in the set.

Let's try an example! Consider the dataset $$\{101,112,124,91,97\}.$$

Good job!

Fun side note: if you're sampling a population and taking a variance of that subset of data, it's called a sample variance. The formula is just slightly different: $$s^2=\sum\limits_{i=1}^{n}\frac{(X_i-\overline{X})^2}{n-1}$$ where $s^2$ is the sample variance. The $n-1$ stems from something called Bessel's correction, which accounts for the bias created by only analyzing a subset of the data. If you're feeling adventurous, try looking into Bessel's correction and the proofs behind why it works!

Variance

In statistics, variance measures how much a set of (random) numbers differ or are "spread out" from from their average value.

In other words, the variance measures how far each number in a data set is from the mean of the data set.

In order to calculate variance, we

• 1. find the average or mean of the data set
• 1. take the differences between each number in the set and the mean
• 1. square each of these differences
• 1. sum up all of the squared differences
• 1. divide this sum by the number of values in the set

And then you have the variance of your set!

(Video) Variance - Clearly Explained (How To Calculate Variance)

by Top Tip Bio

Check out this video by Tip Top Bio to learn more about variance!

As is explained in the video, variance is the average of the squared differences from the mean. Essentially, we want to know how far away each datapoint is from the mean, square these distances, then add these squared values together. We can then divide the resulting number by the degrees of freedom, which is the sample size minus 1.

Variance by itself is hard to analyze because the units are squared, however if we take the square root of the variance, we will get the standard deviation. Both variance and standard deviation help us understand the variability of a dataset.

Variance

Next up in our lessons in statistics is variance! Variance is a measure of how far each item in a dataset is from the mean. In essence, variance measures how "spread out" a dataset is. There are two types of variance: population variance and sample variance. Population variance deals with the entire dataset, while sample variance only deals with a small subset. This may be useful in clinical research, when a researcher may want to only consider variance of a certain subpopulation, like kids, instead of considering all people.

It may seem easier to manually find the variance by hand, but that becomes difficult if there is a high number of data points (i.e., a high sample size). In those instances, we will use the variance formula.

To find the variance, we use this formula:

$$\sigma^2=\sum\limits_{i=1}^{n}\frac{(X_i-\overline{X})^2}{n}$$

, where $\sigma^2$ is the variance, $X_i$ is the $i$th element, $\overline{X}$ is the mean, and $n$ is the size of the dataset.

There are a couple of related concepts. Standard deviation is the square root of variation, which you will learn about in a few lessons.

Related to variance is also squared deviation from the mean (SDM). When considering a theoretical distribution, the variance is the expected value of the SDM. In actual experimental data, variance is the average value of the SDM.