What is the difference between deviation and standard deviation?
In my textbook, it describes the standard deviation "as a number that measures how far data values are from their mean". On the other hand, it says this about deviation: "if x is a number, then the difference "x- mean" is called the deviation". Are they the same or are they different from each other?
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$\begingroup$They're different. The deviation as you have defined it is tied to a single value - how far that particular value is from the mean. The standard deviation, however, actually takes the square root of the average of the squares of these deviations, for every value in the data set!
Let's express this symbolically for clarity.
Let $\mu$ be the mean. Then for a value $x$ in a data set $S$ with $n$ values, $x-\mu$ would be the deviation. But the standard deviation is given by
$$\sigma = \sqrt{\frac{1}{n} \sum_{x\in S} (x-\mu)^2}$$
(Note: At least in my statistics class, we distinguish between the sample standard deviation and that of the population, and in the sample case use $n-1$ in the above formula instead. I'm not exactly sure why we do so, but I think it's important to note. Of course that also means that, for the sample, it's not quite the average of the squares of the deviations, but it's close!)
$\endgroup$ $\begingroup$Deviation, is as you said, how far a single number is from the mean.
However, a standard deviation (describing a set of numbers) is the "root-mean-square" of the deviations. So the standard deviation is basically like the average deviation of the whole sample from the mean.
$\endgroup$ $\begingroup$If $x$ is a sample of a random variable $X$, then the deviation of $x$ from the mean is $x - \mathbb{E}[X]$.
The variance $\mathbb{V}[X]$ of $X$ is the expected value of the square of the deviation, that is, $\mathbb{E}_{x \sim X}[(x-\mathbb{E}[X])^2]$.
The standard deviation $\sigma(X)$ of $X$ is the square root of the variance of $X$. This definition ensures that $\sigma(cX) = c\sigma(X)$, that is, the units of the standard deviation is the same as the units of $X$.
Very roughly speaking, the standard deviation measures the average order of magnitude of the deviation of $X$ from its mean. When $X$ has normal distribution, then the deviation of $X$ from its mean is $\sigma(X)$ times a standard Gaussian. That is, for normally distributed variables, $\sigma(X)$ is the scale of the deviations from the mean.
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