standard deviation on vector quantity?

Question

Do you know definition and a way to calculate?

Answer 1

This is unusual, but I can give you some ideas. The standard deviation is a measure of central tendency usually applied to scalar quantities. A vector, of course, is not a scalar, and it's characterized by both magnitude and direction. So there's the contradiction.

Having said that, though, there are a few approaches you might look at. First, does direction matter? Example: 200 grasshoppers are placed inside a small circular enclosure, and then released. Ten minutes later, the location of each grasshopper is tabulated as a vector quantity, with magnitude and direction.

In this example, direction doesn't matter, and you can find the mean and standard deviation of the distance hopped.

But suppose direction matter. In two dimensions, your vectors can be expressed in terms of magnitude and angle (0 to 360 degrees).

In this case, especially if you have a lot of data, you can do a scatter diagram of magnitude vs. direction, then run a least squares line through it. If there's a relationship between magnitude and direction -- (find the correlation coefficient) -- then you can use the standard error of estimate in the same way you use a standard deviation.

An example of this last might be the alignment of iron filings on a sheet of paper subjected to an electromagnetic field. You'll get strong directional correlation.

If your vectors vary in three dimensions, then I don't think you can do anything like this that's meaningful.

If you want to use the standard error, here's the formula:

S.E. = sqrt{[sum(y^2) - a sum(y) - b sum(xy)] / (n-2)}

where a and b are the regression coefficients:

b = [n sum(xy) - sum(x) sum(y)]/{n sum(x^2) - [sum(x)]^2}

a = ybar - b xbar

and xbar, ybar are the means of x and y respectively.

Answer 2

It's most likely a vector of standard deviations. This

Answer 3

the classic deviation would nicely be used as a level of threat. Or how volitale an funding motor vehicle is. severe favourite deviation = extra threat Low favourite deviation = a lot less threat