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Obviously the meaning of the standard deviation is its relation to the mean, and a standard deviation around a tenth of the mean is unremarkable (e.g. for IQ: SD = 0.15 * M). But what is considered "small" and what is "large", when it comes to the relation between standard deviation and mean?
The standard deviation is one particular measure of the variation. There are several others, Mean Absolute Deviation is fairly popular. The standard deviation is by no means special. What makes it appear special is that the Gaussian distribution is special. As Pointed out in comments Chebyshev's inequality is useful for getting a feeling.
A variable, on the other hand, has a standard deviation all its own, both in the population and in any given sample, and then there's the estimate of that population standard deviation that you can make given the known standard deviation of that variable within a given sample of a given size. So it's important to keep all the references ...
Incidentally we have the median which is 13 times smaller than the mean. For the fast answer, the plot does not exist because the mean and standard deviation are dominated by single outliers separated by more than the mean's value. If I set my histogram based on the median, I lose the important part of the graph off the right.
The higher the standard deviation the more variability or spread you have in your data. Standard deviation measures how much your entire data set differs from the mean. The larger your standard deviation, the more spread or variation in your data. Small standard deviations mean that most of your data is clustered around the mean. In the following graph, the mean is 84.47, the standard ...
In large samples* from a normal distribution, it will usually be approximately the case -- about 99.7% of the data would be within three sample standard deviations of the sample mean (if you were sampling from a normal distribution, your sample should be large enough for that to be approximately true - it looks like there's about a 73% chance ...
The standard deviation does, indeed, give more weight to those farther from the mean, because it is the square root of the average of the squared distances. The reasons for using this (rather than the mean absolute deviation that you propose, or the median absolute deviation, which is used in robust statistics) are partly due to the fact that ...
$\begingroup$ Let's save I have the following set of numbers: {1,3,5} the mean is 3 and the standard deviation is 2. for my understanding, adding another number to the set which is far less than 1 standard deviation from the mean (for example: 4) will decrease the variance. I'm asking whether this rule always applies.
The mean absolute deviation is about .8 times (actually $\sqrt{2/\pi}$) the size of the standard deviation for a normally distributed dataset. Regardless of the distribution, the mean absolute deviation is less than or equal to the standard deviation.
To show how standard deviation affects correlation, we have to use a method that doesn't apply a constant to all the values, but will shift the standard deviation in a way that doesn't completely alter the relationship the original values had. To do so, we can simply double the last two units of x and y, which will change the standard deviation ...