Standard Deviation & Variance

Definition

Standard Deviation

Intuitively, the standard deviation tells us how spread out a set is. Two data sets may have the same mean, but may have a drastically different standard deviation. See Figure 1 for an example.

These two datasets are obviously very different, but just looking at the mean would not tell us that. We need to also consider the standard deviation of the set. Of course, there are hundreds of other values we could extract from these sets to get more information, but standard deviation remains a core metric. ¹

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There are two ways to summarize the numbers: by quantifying their similarities or their differences. Ways of quantifying their similarity to one another are formally called “measures of central tendency”. Those measures include the mean, median and mode. Ways of quantifying their differences are called “measures of variability” and include the variance and standard deviation. The standard deviation should tell us how a set of numbers are different from one another, with respect to the mean. ²

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The standard deviation is the average amount of variability in your dataset. It tells you, on average, how far each value lies from the mean.

A high standard deviation means that values are generally far from the mean, while a low standard deviation indicates that values are clustered close to the mean. ³

Variance

The variance is a measure of variability. It is calculated by taking the average of squared deviations from the mean.

Variance tells you the degree of spread in your data set. The more spread the data, the larger the variance is in relation to the mean. ⁴

Who

TBS

What

TBS

Why

See Theoretical Knowledge Vs Practical Application.

How

Many of the References and Additional Reading websites and Videos will assist you with understanding and calculating the standard deviation and variance.

As some professors say: “It is intuitively obvious to even the most casual observer.“

References

¹ Cole. “What, Exactly, Is Standard Deviation?” 2021. Medium. https://www.cantorsparadise.com/what-exactly-is-standard-deviation-5a202c3f8c26.

² Alhazmi, Fahd. “A Visual Interpretation Of The Standard Deviation”. 2020. Medium. https://towardsdatascience.com/a-visual-interpretation-of-the-standard-deviation-30f4676c291c.

³ Bhandari, Pritha. “How To Calculate Standard Deviation (Guide) | Formulas & Examples”. 2020. Scribbr. https://www.scribbr.com/statistics/standard-deviation/.

⁴ Bhandari, Pritha. “What Is Variance? | Definition, Examples & Formulas”. 2020. Scribbr. https://www.scribbr.com/statistics/variance/.

Additional Reading

Jamison, Mark. “Why Do We Use The Standard Deviation?”. 2022. Medium. https://towardsdatascience.com/why-do-we-use-the-standard-deviation-51d5d1a502a8.

Konstantakos, Vasileios. “Standard Deviation Vs. Variance: What’S The Difference?”. 2021. Medium. https://medium.com/@vkonstantakos/standard-deviation-vs-variance-whats-the-difference-b9c671d6f5cf.

Pelzel, Kristi. “Data Visualization Part 6: Mean, Variance, And Standard Deviation”. 2021. Medium. https://medium.com/upskilling/data-visualization-part-6-mean-variance-and-standard-deviation-912f4045081d.

The post will focus on calculating mean, variance, and standard deviation. We will also skim over continuous distributions and the use and limitations of p-values in hypothesis testing.

I’ll explain mean by using this story example, then explain variance and standard deviation by using baseball batting average as the example.

I’ll explain continuous distributions with an example of “height” (as a single number representing a continuous set of values).

Then, I’ll talk about hypothesis testing and p-values with an example.

“Standard Deviation And Variance”. 2022. mathsisfun.com. https://www.mathsisfun.com/data/standard-deviation.html.

“What Are The 4 Main Measures Of Variability?” 2022. Scribbr. https://www.scribbr.com/frequently-asked-questions/what-are-the-4-main-measures-of-variability/.

Videos

This statistics video tutorial explains how to use the standard deviation formula to calculate the population standard deviation. The formula for the sample standard deviation is also provided. This video shows you the variables associated with the sample mean and the population mean. In addition, it discusses how to calculate variance from standard deviation.

The most common measures of dispersion for metric variables are the standard deviation and the variance in statistics. These two measures relate each expression of a variable to the mean and indicate how much the individual expressions scatter around the mean. What is the standard deviation and how do I calculate it? In statistics, the standard deviation gives you the spread of a variable around its mean. The standard deviation, is the average distance of all measured values of a variable from the mean of the distribution. The measure of dispersion standard deviation thus indicates how much the individual values scatter around the mean value. If the individual values scatter strongly around the mean, the result is a large standard deviation of the variable. Sorry, in the example of course a standard deviation of 11.50 comes out and not 12.06!!!! – DATAtab

In statistics, variance measures the deviation from the mean. To calculate the variance, the sum of the squared variances is divided by the number of values. Variance The variance now describes the squared average distance from the mean. Because the values are squared, the result has a different unit (just the unit squared) than the original values. Therefore, it is difficult to relate the results. Difference between variance and standard deviation So, the difference between the variance and standard deviation dispersion parameter is that the standard deviation measures the average distance from the mean and the variance measures the squared average distance from the mean. Put another way, the variance is the squared standard deviation and the standard deviation is the root of the variance. – DATAtab

In this lesson, you’ll learn about the concept of variance in statistics. We’ll discuss how variance is derived and what the equations of variance means. We will also cover how variance is very closely related to standard deviation in statistics.