Standard Deviation vs. Variance: What’s the Difference?
1 Introduction
The measures of central tendency are not adequate to fully describe data. For instance, two datasets following a normal distribution can have the same mean value but still be entirely different (Fig. 1). Thus, one needs to know the extent of variability for a given dataset. This variability can be captured by measures of dispersion. A measure of statistical dispersion is a non-negative real number that describes the extent to which a distribution is stretched or squeezed. Variance and standard deviation (SD) are two measures that are commonly used. We discuss their similarities and differences in the following section.
A measure of statistical dispersion is a non-negative real number that describes the extent to which a distribution is stretched or squeezed.
2 Theoretical framework
The most important quantity associated with a random variable X, other than the mean, is its variance, which is denoted by Var[X] and is defined as the expected value of the squared deviation from its mean μ =E[X]:
For example, if X is a numerical discrete random variable with distribution p(x) and expected value μ = E[X], the variance of X is:
In particular, for a set of data {x₁, x₂, …, xₙ} for a population of size n, we define the population variance, denoted by σ² to be the average squared distance from the mean, μ:
Note that from the definition, the variance is always non-negative, and if the variance is equal to zero, then the random variable X takes a single constant value, which is its expected value μ.
On the other hand, another measure of dispersion is the standard deviation (SD) of X, which is defined as the square root of the variance and is denoted by σ:
Although variance and standard deviation are quite similar, they have very different purposes. Specifically, the SD is usually more useful to describe the variability of the data while the variance is usually much more useful mathematically. For example, the sum of uncorrelated distributions (random variables) has a variance that is the sum of the variances of those distributions. Formally, variances are additive; it holds that for independent random variables X₁, …, Xₙ:
A similar expression exists in the general - without independence - case with a correction using covariance terms. However, this is not true for the SD. In general, the square root transformation complicates the required calculations and makes standard deviation more difficult to work with analytically.
On the other hand, the SD has the convenience of being expressed in units of the original variable. Thus, it is often easier to interpret.
For example, if X measures length in meters, the units of variance are square meters, while the units of the standard deviation are meters.
An additional reason why SD is a very useful measure of dispersion is that, if the observations are from a normal distribution, then 68% of observations lie between mean μ ±1 SD, 95% of observations lie between mean μ ± 2 SD and 99.7% of observations lie between μ ± 3 SD.
Thus, since we can often approximate the distribution of X by a normal distribution, we can further describe it in terms of SD (e.g., “X is likely to be within ± 2 standard deviations of the mean”).
The SD is usually more useful to describe the variability of the data while the variance is usually much more useful mathematically.
Finally, we briefly mention some key points to further illustrate the differences between these two measures of dispersion:
- The mean and variance are the two parameters that determine a normal distribution.
- The Chebyshev inequality bounds the probability of an observed random variable being within k standard deviations of the mean.
- The standard deviation is used to normalize statistics for statistical tests (e.g., the known SD is used to normalize a sample mean for the z-test, or the sample SD is used to normalize the sample mean when the true SD is unknown, resulting in the t-test).
3 Conclusion
In summary, measures of central tendency when combined with measures of dispersion can more clearly describe the data of our choice. However, it is crucial to use the appropriate measure for our analysis to correctly interpret our results. As we briefly illustrated, this is not always an easy task. The standard deviation - while being the square root of the variance - has some unique properties and captures a different aspect of our data. Therefore, we should consider this when designing experiments and presenting their results.
