I feel like people might be overcomplicating this. If you take a sample from a population, you get two main statistics from it: The mean, and the deviation. One describes the center of the data, the other the distribution around it. Imagine you kept drawing new samples again and again. You can make a list of the means, right? They should all be fairly close, but the random sampling means they're all slightly different. That list of means has it's own mean - and it's own deviation. That deviation is the standard error of the mean. It's a measure of the distribution of means in many samples of the same population. Now, the formula you're probably familiar with obviously doesn't draw many samples from the population! It's an estimate of the SEM, not the actual SEM. It uses a single sample deviation and the number of elements in that sample to make the estimate. More on reddit.com

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. More on stats.stackexchange.com

The variance is the average squared distance from the mean. It has lots of nice statistical properties, but it's hard to interpret because it's squared. Standard deviation is the square root of the variance. It loses a lot of the nice properties of the variance, but getting rid of the square gives a good scale for talking about how far data points are from the mean. For instance, you could discuss outliers that are more than 5 standard deviations for the mean (i.e. mean + 5 times the standard deviation), or discuss the proportion of observations that are within 1 standard deviation of the mean (in either direction). IQ tests are designed to have a mean of 100 and a standard deviation of 15 so that it's easy to know that a score of 130 is 2 standard deviations above the mean (and thus quite good!). The "standard deviation of the mean" (commonly referred to as the standard error) is what you nicely described as the "average result plus or minus error". It's on the same scale as the standard deviation (i.e. not squared), but instead of describing the variability in the individual data points, it describes the variability in the mean based on how many data points you've looked at. So with the IQ test, if you test an entire school and the average score is 103 instead of 100, the standard error is what's going to help you figure out if the school is actually above average or if the difference between 100 and 103 is just random "error". More on reddit.com

This is going to be a long answer, so please bear with me.

Let's imagine you want to know the average height of a population. To do so, you take 10 people and measure each of them. Your values, in inches, are:

60, 62, 62, 64, 65, 66, 66, 67, 70, 72

Your mean is obviously 65.4

Cool. Now let's say you want to know how different each of your sample persons is. You would use Measures of Dispersion, which are standard deviation, standard error, and variance.

Standard deviation describes the average difference of the data compared to the mean. It is simply the average amount each of the data points differs from the mean. So 60 is 5.4 inches from the mean. 62 is 3.4 inches from the mean. So on and so forth. You just add all these numbers up and take their average. You know now the average difference of each of the data points compared to the mean.

Now you know the average difference, but let's pretend you just want to know how different all the data is from each other. This is the variance. Whatever text you're reading probably tells you how to calculate it, so I won't bore you with those details. Essentially, the variance tells you the "spread" of the data. A dart board in which the darts are very far apart has more variance than a board in which everything hit the bull's eye.

Finally, imagine you weren't satisfied with this one sample. You decide to take a second sample. Obviously, you're going to get different people, so your mean might be a little different. This time it might be 67. If you took a bunch of different samples you would get a bunch of different means and you could plot them all on a graph. If you took the standard deviations of THESE values, you would have the standard error of the mean. It's a measure of the average error of your sample means.

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