Standard Deviation: Formula (8) Approximation Explained

In summary, the formula (8) on this page is approximately the standard deviation of a function when its variances are small.
  • #1
daudaudaudau
302
0
Hi all.

If I have a function V(x1,x2,x3,x4) and I want to calculate it's standard deviation when x1,x2,x3,x4 are standard normal and their variances are small, then formula (8) on this page
http://www.devicelink.com/mem/archive/99/09/003.html" is an approximation to the standard deviation. Can anyone offer me a proof or tell me where I can read more about this formula?

-Anders
 
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  • #3
Hi.

Hmm, I think it's the clever part I don't understand then. Must I taylor expand V and then compute the standard deviation? And also, I can't see the connection between "difference" and "variance". I'd appreciate a few formulas. Thanks.

-Anders
 
  • #4
I thought it would follow from dV = Sum[(partial V/partial xi) dxi, i=1,2,3,4]. The "d" operator is similar to "deviation" (e.g., from the mean). Let's say you have only 2 x's. Then dv = (Dv/Dx1) dx1 + (Dv/Dx2) dx2 ==> dv^2 = (Dv/Dx1)^2 dx1^2 + (Dv/Dx2)^2 dx2^2 + ignored term* approx. equal to (Dv/Dx1)^2 dx1^2 + (Dv/Dx2)^2 dx2^2. (I've used capital D for "partial.")

In fact, you don't even need the link I posted.

*This is the interaction term 2(Dv/Dx1)(Dv/Dx2) dx1 dx2. If you think that for a given "random draw" either of dx1 or dx2 (but not necessarily both, and you don't know which) is likely to be "very small" then you can assume 2(Dv/Dx1)(Dv/Dx2) dx1 dx2 = 0.
 
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  • #5
Okay. The only part I don't understand now is why dx is the same as a standard deviation ?
 

1. What is the formula for standard deviation?

The formula for standard deviation is the square root of the variance. The variance is calculated by finding the average of the squared differences between each data point and the mean.

2. How is standard deviation used in statistics?

Standard deviation is used as a measure of how spread out the data is in a dataset. It helps to understand the variability of the data and how far away the data points are from the mean.

3. How is standard deviation related to the normal distribution?

The normal distribution, also known as the bell curve, is a commonly used statistical distribution that is based on the mean and standard deviation of a dataset. The standard deviation determines the shape and spread of the curve, with 68% of the data falling within one standard deviation of the mean.

4. What is the difference between population and sample standard deviation?

Population standard deviation is used when calculating the standard deviation of a dataset that includes all possible data points. Sample standard deviation is used when calculating the standard deviation from a smaller subset of the population. Sample standard deviation tends to be slightly higher than population standard deviation.

5. How is standard deviation approximated?

Standard deviation can be approximated using a shortcut formula called the "empirical rule". This rule states that approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. This approximation is only accurate for datasets that follow a normal distribution.

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