Difference between standard error and standard deviation I'm struggling to understand the difference between the standard error and the standard deviation. How are they different and why do you need to measure the standard error?
 A: (note that I'm focusing on standard error of the mean, which I believe the questioner was as well, but you can generate a standard error for any sample statistic)
The standard error is related to the standard deviation but they are not the same thing and increasing sample size does not make them closer together. Rather, it makes them farther apart. The standard deviation of the sample becomes closer to the population standard deviation as sample size increases but not the standard error.
Sometimes the terminology around this is a bit thick to get through.
When you gather a sample and calculate the standard deviation of that sample, as the sample grows in size the estimate of the standard deviation gets more and more accurate. It seems from your question that was what you were thinking about. But also consider that the mean of the sample tends to be closer to the population mean on average. That's critical for understanding the standard error.
The standard error is about what would happen if you got multiple samples of a given size. If you take a sample of 10 you can get some estimate of the mean. Then you take another sample of 10 and new mean estimate, and so on. The standard deviation of the means of those samples is the standard error. Given that you posed your question you can probably see now that if the N is high then the standard error is smaller because the means of samples will be less likely to deviate much from the true value.
To some that sounds kind of miraculous given that you've calculated this from one sample. So, what you could do is bootstrap a standard error through simulation to demonstrate the relationship. In R that would look like:
# the size of a sample
n <- 10
# set true mean and standard deviation values
m <- 50
s <- 100

# now generate lots and lots of samples with mean m and standard deviation s
# and get the means of those samples. Save them in y.
y <- replicate( 10000, mean( rnorm(n, m, s) ) )
# standard deviation of those means
sd(y)
# calcuation of theoretical standard error
s / sqrt(n)

You'll find that those last two commands generate the same number (approximately). You can vary the n, m, and s values and they'll always come out pretty close to each other.
A: Here is a more practical (and not mathematical) answer:


*

*The SD (standard deviation) quantifies scatter — how much the values vary from one
another.  

*The SEM (standard error of the mean) quantifies how precisely you know the true mean of
the population. It takes into account both the value of the SD and
the sample size.  

*Both SD and SEM are in the same units -- the units of the data.

*The SEM, by definition, is always smaller than the
SD. 

*The SEM gets smaller as your samples get larger. This makes
sense, because the mean of a large sample is likely to be closer to
the true population mean than is the mean of a small sample. With a
huge sample, you'll know the value of the mean with a lot of
precision even if the data are very scattered. 

*The SD does not change
predictably as you acquire more data. The SD you compute from a
sample is the best possible estimate of the SD of the overall
population. As you collect more data, you'll assess the SD of the
population with more precision. But you can't predict whether the SD
from a larger sample will be bigger or smaller than the SD from a
small sample. (This is a simplification, not quite true. See comments below.)


Note that standard errors can be computed for almost any parameter you compute from data, not just the mean. The phrase "the standard error" is a bit ambiguous. The points above refer only to the standard error of the mean.
(From the GraphPad Statistics Guide that I wrote.)
A: Let $\theta$ be your parameter of interest for which you want to make inference. To do this, you have available to you a sample of observations $\mathbf{x} = \{x_1, \ldots, x_n \}$ along with some technique to obtain an estimate of $\theta$, $\hat{\theta}(\mathbf{x})$. In this notation, I have made explicit that $\hat{\theta}(\mathbf{x})$ depends on $\mathbf{x}$. Indeed, if you had had another sample, $\tilde{\mathbf{x}}$, you would have ended up with another estimate, $\hat{\theta}(\tilde{\mathbf{x}})$. This makes $\hat{\theta}(\mathbf{x})$ a realisation of a random variable which I denote $\hat{\theta}$. This random variable is called an estimator. The standard error of $\hat{\theta}(\mathbf{x})$ (=estimate) is the standard deviation of $\hat{\theta}$ (=random variable). It contains the information on how confident you are about your estimate. If it is large, it means that you could have obtained a totally different estimate if you had drawn another sample. The standard error is used to construct confidence intervals. 
A: To complete the answer to the question, Ocram nicely addressed standard error but did not contrast it to standard deviation and did not mention the dependence on sample size.  As a special case for the estimator consider the sample mean.  The standard error for the mean is $\sigma \, / \, \sqrt{n}$ where $\sigma$ is the population standard deviation.  So in this example we see explicitly how the standard error decreases with increasing sample size. The standard deviation is most often used to refer to the individual observations. So standard deviation describes the variability of the individual observations while standard error shows the variability of the estimator. Good estimators are consistent which means that they converge to the true parameter value. When their standard error decreases to 0 as the sample size increases the estimators are consistent which in most cases happens because the standard error goes to 0 as we see explicitly with the sample mean.
