Standard deviation why square

Example: squares can be integrated, differentiated, can be used in trigonometric, logarithmic and other functions, with ease. Naturally you can describe dispersion of a distribution in any way meaningful absolute deviation, quantiles, etc.

One nice fact is that the variance is the second central moment, and every distribution is uniquely described by its moments if they exist. Another nice fact is that the variance is much more tractable mathematically than any comparable metric. Another fact is that the variance is one of two parameters of the normal distribution for the usual parametrization, and the normal distribution only has 2 non-zero central moments which are those two very parameters.

Even for non-normal distributions it can be helpful to think in a normal framework. As I see it, the reason the standard deviation exists as such is that in applications the square-root of the variance regularly appears such as to standardize a random varianble , which necessitated a name for it.

A different and perhaps more intuitive approach is when you think about linear regression vs. In other words, whether to use absolute or squared error depends on whether you want to model the expected value or the median value. This is an old thread, but most answers focus on analytical simplicity, which IMO is a weak argument in times of computers although numerical stability might be an issue when using absolute values in optimization routines.

Here are some more fundamental arguments in favor of the variance. My guess is this: Most populations distributions tend to congregate around the mean.

The farther a value is from the mean, the rarer it is. In order to adequately express how "out of line" a value is, it is necessary to take into account both its distance from the mean and its normally speaking rareness of occurrence. Squaring the difference from the mean does this, as compared to values which have smaller deviations. Once all the variances are averaged, then it is OK to take the square root, which returns the units to their original dimensions.

If your sample has values that are all over the chart then to bring the Some say that it is to simplify calculations.

Using the positive square root of the square would have solved that so that argument doesn't float. Sign up to join this community.

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Create a free Team What is Teams? Learn more. Why square the difference instead of taking the absolute value in standard deviation? Ask Question.

Asked 11 years, 3 months ago. Active 7 days ago. Viewed k times. Improve this question. Isn't it like asking why principal component are "principal" and not secondary? They focus on ease of mathematical calculations which is nice but by no means fundamental or on properties of the Gaussian Normal distribution and OLS.

Around Gauss started with least squares and variance and from those derived the Normal distribution--there's the circularity. A truly fundamental reason that has not been invoked in any answer yet is the unique role played by the variance in the Central Limit Theorem.

Another is the importance in decision theory of minimizing quadratic loss. Show 21 more comments. Active Oldest Votes. The benefits of squaring include: Squaring always gives a positive value, so the sum will not be zero.

Squaring emphasizes larger differences—a feature that turns out to be both good and bad think of the effect outliers have. Improve this answer. Pitouille 1, 4 4 silver badges 16 16 bronze badges. Tony Breyal Tony Breyal 3, 1 1 gold badge 17 17 silver badges 13 13 bronze badges. I wasn't implying that anything about absolute values in that statement. The size in each dimension is the difference from the mean for that sample. Show 5 more comments. Rich Rich 4, 1 1 gold badge 20 20 silver badges 20 20 bronze badges.

This makes analytical optimization more difficult. Consider the 1 dimension case; you can express the minimizer of the squared error by the mean: O n operations and closed form. You can express the value of the absolute error minimizer by the median, but there's not a closed-form solution that tells you what the median value is; it requires a sort to find, which is something like O n log n.

Least squares solutions tend to be a simple plug-and-chug type operation, absolute value solutions usually require more work to find. Median does not require sorting. Show 3 more comments.

Reed Copsey Reed Copsey 1, 8 8 silver badges 5 5 bronze badges. It's essentially a Pythagorean equation. Add a comment. Neil G Neil G Tools for Fundamental Analysis. Portfolio Management. Advanced Technical Analysis Concepts. Risk Management. Your Privacy Rights. To change or withdraw your consent choices for Investopedia. At any time, you can update your settings through the "EU Privacy" link at the bottom of any page.

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Popular Courses. Financial Analysis How to Value a Company. Key Takeaways Standard deviation looks at how spread out a group of numbers is from the mean, by looking at the square root of the variance. The variance measures the average degree to which each point differs from the mean—the average of all data points.

The two concepts are useful and significant for traders, who use them to measure market volatility. This topic has 32 replies, 10 voices, and was last updated 3 years, 4 months ago by Mike Carnell.

Viewing 33 posts - 1 through 33 of 33 total. September 14, at pm So, in the method for calculating the standard deviation at one point we subtract the Mean. The mean is the average data point value within a data set.

Data are factual information used as a basis for reasoning, discussion or calculation; often this term refers to quantitative information. September 15, at am Because we are subtracting from Mean.

September 15, at pm September 17, at pm September 20, at pm May 2, at pm May 3, at am Amit Kumar Ojha Participant. AmitOjha Include AmitOjha in your post and this person will be notified via email. May 4, at am February 26, at pm February 27, at am Mike-Carnell Include Mike-Carnell in your post and this person will be notified via email. That is a Lot. A collection of individual pieces from a common source, possessing a common set of quality characteristics and submitted as a group for acceptance at one time.

Andy-Parr Include Andy-Parr in your post and this person will be notified via email. That looks good and is the Mean Deviation , but what about this case:. That is nice! The Standard Deviation is bigger when the differences are more spread out In fact this method is a similar idea to distance between points , just applied in a different way. And it is easier to use algebra on squares and square roots than absolute values, which makes the standard deviation easy to use in other areas of mathematics.

The average of the squared differences from the Mean.