# Understanding the Differences Between Variance and Standard Deviation

Do you find statistics and data analysis a bit overwhelming? Maybe you’ve heard about variance and standard deviation, but you aren’t quite sure what they are or how they differ.

Variance and Standard Deviation are the two most fundamental terms in statistics and are important for analyzing data. Variance measures the dispersion of data, whereas the standard deviation measures the variation of data from the mean. Standard deviation is more interpretable than a variance and is considered a better measure for interpreting dispersion.

In this blog post, I will discuss the differences between variance and standard deviation. By the end of this article, you will understand not only the differences between variance and standard deviation but also their applications and importance.

If you wonder what are discrete and continuous variables, how they are different, and why it matters? I wrote a whole article that I suggest you read.

## What Is Variance?

Variance is a statistical term that is used to measure the dispersion of data. It is calculated by taking the squared differences between each data point and the mean and then taking the average of those squared differences.

Unlike range and interquartile range (IQR), variance is a measure of dispersion that considers the spread of all data points in a data set. Variance and the standard deviation (simply the square root of the variance) are the measures of dispersion most often used.

For example, if we have four data points, 2, 4, 6, 8, and the mean is 5, then the variance can be calculated by taking each point’s difference from the mean, squaring it, and then averaging all four values. The variance is 6.661. This essentially means that the data points are dispersed around the mean by 2.581 on average.

## What Is Standard deviation?

Standard deviation measures the amount of variation or dispersion of data from the mean. It is the square root of the variance, which signifies the spread of data points from the mean.

Standard deviation is helpful when comparing the spread of two different data sets with approximately the same mean. In this case, generally, the data set with the smaller standard deviation has a narrower spread of measurements around the mean and typically has comparatively fewer high or low values.

An item selected randomly from a data set with a low standard deviation will have a better chance of being near the mean than an item with a higher standard deviation. However, the standard deviation is affected by extreme values. A single extreme value can have a big impact on the standard deviation.

In simple words, standard deviation measures the uncertainty or the error from the mean. It is widely used in various data analysis methods to identify patterns or trends in the dataset.

### Important Properties When Using Standard Deviation:

• Standard deviation is very sensitive to extreme values. For instance, a single very extreme value can generally increase the standard deviation and lead to misrepresenting the dispersion.
• For two data sets with the same mean, the data set with the bigger standard deviation is where the data is more spread out from the center.
• The standard deviation is equal to 0 if all values are equal (because all values are equal to the mean).

## Why Standard Deviation Is Most Popular?

Standard deviation is a popular and important measure of dispersion because of its relation with the normal distribution, which is generally used to describe most natural phenomena. When a variable follows a normal distribution (or normally distributed), the histogram is symmetric around the mean and bell-shaped, and the most suitable measures of central tendency and dispersion are the standard deviation and the mean (Source: Statistics Canada)

It is an essential probability distribution and reasonably easy to use. Confidence intervals are typically based on the standard normal distribution.

## What Does Standard Deviation Tell You?

A standard deviation (or σ) measures how dispersed the data is in relation to the mean. A low standard deviation indicates data are clustered around the mean. On the other hand, a high standard deviation suggests data are more spread out (Source: National Library of Medicine)

A standard deviation close to zero means that data points are close to the mean. In contrast, a high standard deviation indicates data points are above, and a low standard deviation means data points are below the mean.

For example, a sample with a standard deviation equal to 5 suggests that, on average, the distance between each data point in an entire dataset differs from the mean by a value of 5.

When Should You Not Use The Standard Deviation:

If you encounter any of the below situations, do not use the standard deviation; instead, it is better to use the interquartile range:

• The data set is too small.
• The distribution is skewed (asymmetric)
• The data set contains extreme values.

## Difference Between Variance And Standard Deviation

To start with, variance and standard deviation are both measures of how spread out a set of data is. Variance is the average of the squared differences from the mean, while the standard deviation is the square root of the variance.

The major difference between variance and standard deviation is that variance is in squared units while standard deviation is in the same units as the data.

Imagine that you have data on the heights of ten people. The average height is 170 cm, and the variance is 30 cm squared. The standard deviation is the square root of the variance or, in this case, approximately 5.5 cm. This means the height of each person varies up to 5.5 cm from the average.

The variance is an indicator of the variability in the heights, while the standard deviation tells us about the average distance between each height and the mean height.

Generally, variance is not the easiest measure of variability to interpret because the variance of a dataset is not in the same units as the data itself. And since the standard deviation is a measure of dispersion that is more interpretable as it takes into account the units of the data.

In data analysis cases, where meaningful communication of dispersion is required with a clear understanding of how spread out the dataset is, standard deviation is a better option.

Another difference between variance and standard deviation is that the square of the standard deviation is defined as the variance. Therefore, both terms are related, and one can be derived from another. Whenever you calculate the standard deviation of a dataset, you are essentially calculating the square root of the variance.

It is also worth noting that variance and standard deviation are used in different applications. Variance is more commonly used in probability, theoretical statistics, and inferential statistics. In contrast, the standard deviation is used in descriptive statistics, such as summarizing data in graphs or charts and calculating confidence intervals of mean estimations.

If you want to learn more about the concepts of variance and standard deviation, I encourage you to watch this excellent video from Khan Academy explaining the differences between variance and standard deviation.

### Wrapping Up

Variance and standard deviation are both important statistical tools that measure the spread of a set of data. Variance measures the variability of the data, while standard deviation measures the average distance between each data point and the mean.

While the variance is used in theoretical and inferential statistics, the standard deviation is used in descriptive statistics, including graphical representations of data.

I believe that understanding the difference between variance and standard deviation is crucial, as they provide insights into how data is distributed and can assist in making informed decisions in various fields, including business, finance, and science.

Altiné

I am Altiné. I am the guy behind mathodics.com. When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting mathodics.com.