In statistics, we are often interested in understanding how “spread out” values are in a dataset. To measure this, we often use the following measures of dispersion:
- The range: the difference between the largest and smallest value in a dataset.
- The interquartile range: the difference between the first quartile and the third quartile in a dataset (quartiles are simply values that split up a dataset into four equal parts).
- The standard deviation: a way to measure the typical distance that values are from the mean.
- The variance: the standard deviation squared.
Out of these four measures, the variance tends to be the one that is the hardest to understand intuitively. This post aims to provide a simple explanation of the variance.
Understanding Standard Deviation
Before we can understand the variance, we first need to understand the standard deviation, typically denoted as σ.
The formula to calculate the standard deviation is:
σ = √(Σ (xi – μ)2 / N)
where μ is the population mean, xi is the ith element from the population, N is the population size, and Σ is just a fancy symbol that means “sum.”
In practice, you will rarely need to calculate the standard deviation by hand; instead, you can use statistical software or a calculator.
At its most basic level, the standard deviation tells us how spread out the data values are in a dataset. To illustrate this, consider the following three datasets along with their corresponding standard deviations:
[5, 5, 5] standard deviation = 0 (no spread at all)
[3, 5, 7] standard deviation = 1.63 (some spread)
[1, 5, 99] standard deviation = 45.28 (a lot of spread)
The term “standard deviation” can be understood by looking at the two words that make it up:
- “deviation” – this refers to the distance from the mean.
- “standard” – this refers to the “standard” or “typical”distance that a value is from the mean.
Once you understand standard deviation, it’s much easier to understand variance.
Understanding Variance
The variance, typically denoted as σ2, is simply the standard deviation squared. The formula to find the variance of a dataset is:
σ2 = Σ (xi – μ)2 / N
where μ is the population mean, xi is the ith element from the population, N is the population size, and Σ is just a fancy symbol that means “sum.”
So, if the standard deviation of a dataset is 8, then the variation would be 82 = 64.
Or, if the standard deviation of a dataset is 10, then the variation would be 102 = 100.
Or, if the standard deviation of a dataset is 3.7, then the variation would be 3.72 = 13.69.
The more spread out the values are in a dataset, the higher the variance. To illustrate this, consider the following three datasets along with their corresponding variances:
[5, 5, 5] variance = 0 (no spread at all)
[3, 5, 7] variance = 2.67 (some spread)
[1, 5, 99] variance = 2,050.67 (a lot of spread)
When Would You use Variance Instead of Standard Deviation?
After reading the above explanations for standard deviation and variance, you might be wondering when you would ever use the variance instead of the standard deviation to describe a dataset.
After all, the standard deviation tells us the average distance that a value lies from the mean while the variance tells us the square of this value. It would seem that the standard deviation is much easier to understand and interpret.
In reality, you will almost always use the standard deviation to describe how spread out the values are in a dataset.
However, the variance can be useful when you’re using a technique like ANOVA or Regression and you’re trying to explain the total variance in a model due to specific factors.
For example, you might want to understand how much variance in test scores can be explained by IQ and how much variance can be explained by hours studied.
If 36% of the variation is due to IQ and 64% is due to hours studied, that’s easy to understand. But if we use the standard deviations of 6 and 8, that’s much less intuitive and doesn’t make much sense in the context of the problem.
Another case in which the variance may be better to use than the standard deviation is when you’re doing theoretical statistical work.
In this case, it’s much easier to use the variance when doing calculations since you don’t have to use a square root sign.
Additional Resources
The following tutorials provide additional information about variance:
Sample Variance vs. Population Variance: What’s the Difference?
How to Calculate Sample & Population Variance in Excel