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Sorry for the rudimentary question, but I just want to make sure I understand everything well conceptually. I understand how we get the standard deviation of a population. My questions are as follows:

  1. If we want to describe the spread of a sample of data, why would we not use the same formula we would for the population? In other words, given a population of 20 individuals and a sample of 20 individuals, why don't we divide by N for both of the data sets to express how far on average each data point is from each sample's mean? Given all data points are the same, wouldn't these two necessarily have the same spread, and therefore should have the same numerical value for a measure of spread (standard deviation)?

  2. Can one use the sample standard deviation to estimate the population standard deviation? Is this when the N vs. n-1 question comes into play?

  3. Standard error tells us how far, on average, a given sample mean deviates from the true mean of these means (which will be the population mean), correct?

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    $\begingroup$ Please check the site for possible duplicates. I think you will find that this has already been answered. $\endgroup$ – Michael R. Chernick Nov 28 '19 at 20:37
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  1. You rarely have entire population data at hand. That is why random samples are used to approximate population characteristics. Using population-based formulae with your sample without a correction will yield biased estimates.

  2. N-1 is simply a correction applied to take into account the fact that you are working with a sample, rather than the population itself

  3. Yes (that's called an empirical standard error)

Important note: I think you will gain further insight into the terms and properties of standard error, standard deviation, and population corrections by reading the following threads

  1. Difference between standard error and standard deviation

  2. Intuition behind applying N-1 correction for sample variance

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