Difference between standard error and standard deviation

Question

I'm a beginner in statistics. 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 standard error? Can anyone enlighten me?

Answer 1

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 σ/√n where σ 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 staandard error decreases to 0 as the sample size increases the estimators are consistent which in most cases happens becuase the standard error goes to 0 as we see explicitly with the sample mean.

Answer 2

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.

Answer 3

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.