# How exactly can I get sampling distribution of means for calculation of standard error?

Let's denote population by $$N$$ and sample with which I want to make inference about $$N$$ denote by $$n$$.

Let's assume I have already collected $$n$$. Now I want to get means for sampling distribution of means for calculation of Standard Error for $$n$$.

Question: From where I have to draw multiple random samples for the sampling distribution of means? I need to draw multiple small random samples from $$n$$ or additional random samples (that are not related to $$n$$) from $$N$$?

There are numerous ways of estimating the SE of the mean from a population. It is possible to use either approach you outlined though neither are particularly common. I will describe a few approaches that can be taken below.

In these examples, I will be considering a population $$P$$ of size $$N$$ whose members have some characteristic $$c$$. Let the mean of $$c$$ for this population be denoted $$\mu$$ and the standard deviation of $$c$$ be denoted $$\sigma$$. The standard error of the mean considered is the standard deviation of the sampling distribution containing all samples of size $$n$$ from $$P$$

# Estimate the SE through multiple samples on the population

The most direct approach to estimating the SE is to take a number of samples of size $$n$$ from $$P$$. Each sample can then be used to produce an estimate of the $$\mu$$. The standard deviation of these estimates being the estimate of the standard error.

# Estimate the SE through bootstrapping

The above approach is often considered overly burdensome as it involves repeating the sampling process many times in order to produce an estimate of the SE. To get around this one way is to simulate the process of collecting additional samples, rather than actually collecting them.

The most common method for doing this would be bootstrapping, this process is based on the idea that if a sample $$S$$ of size $$n$$ is taken from population $$P$$, then $$S$$ should be broadly representative of $$P$$. If $$n$$ << $$N$$, any samples (taken with replacement) taken from $$S$$, should also be broadly representative of new samples taken from $$P$$.

By resampling from the original sample in this way multiple estimates of $$\mu$$ can be produced from a single sample.

# Estimating the SE algebraically.

In practice, though both of these methods are quite rare, as in general the SE is estimated using the formula

$$SE = \frac {s }{\sqrt {n}}$$

where $$s$$ is the sample standard deviation.

A derivation of this formula can be found on Wikipedia.

• What are the advantages and disadvantages of bootstrapping method and "Estimating the SE algebraically" method and when to use which of them? – vasili111 Apr 16 at 1:23
• Generally you would want to use the formula (possibly with a finite population correction). Any bootstrapping approach will be less accurate as it is introducing error from the simulation process. bootstrapping is normally used when there is specific cause for example in order to estimate the SE of different sampling strategies compared to what was carried out originally (i.e. a stratified sample compared to a simple random sample). – Ryan Apr 16 at 13:32