Fundamentals of Statistics 3: Sampling :: The standard error of the mean
We saw with the sampling distribution of the mean that every sample we take to estimate the unknown population parameter will overestimate or underestimate the mean by some amount. But what's interesting is that the distribution of all these sample means will itself be normally distributed, even if the population is not normally distributed. The central limit theorem states that the mean of the sampling distribution of the mean will be the unknown population mean. The standard deviation of the sampling distribution of the mean is called the standard error.  In fact, it is just another standard deviation, we just call it the standard error so we know we're talking about the standard deviation of the sample means instead of the standard deviation of the raw data. The standard deviation of data is the average distance values are from the mean.

Ok, so, the variability of the sample means is called the standard error of the mean or the standard deviation of the mean (these terms will be used interchangeably since they mean the same thing) and it looks like this.

Standard Error of the Mean (SEM) = The symbol σ sigma represents the population standard deviation and n is the sample size. Population parameters are symbolized using Greek symbols and we almost never know the population parameters. That is also the case with the standard error. Just like we estimated the population standard deviation using the sample standard deviation, we can estimate the population standard error using the sample standard deviation.

When we repeatedly sample from a population, the mean of each sample will vary far less than any individual value. For example, when we take random samples of women's heights, while any individual height will vary by as much as 12 inches (a woman who is 5'10 and one who is 4'10), the mean will only vary by a few inches. The distribution of sample means varies far less than the individual values in a sample.If we know the population mean height of women is 65 inches then it would be extremely rare to have a sampe mean of 30 women at 74 inches.

In fact, if we took a sample of 30 women and found an average height of 6'1, then we would wonder whether these were really from the total population of women. Perhaps it was a population of Olympic Volleyball players. It is possible that a random sample of women from the general population could be 6'1 but it is extremely rare (like winning the lottery).

The standard deviation tells us how much variation we can expect in a population. We know from the empirical rule that 95% of values will fall within 2 standard deviations of the mean. Since the standard error is just the standard deviation of the distribution of sample mean, we can also use this rule.

So how much variation in the standard error of the mean should we expect from chance alone?  Using the empirical rule we'd expect 68% of our sample means to fall within 1 standard error of the true unknown population mean. 95% would fall within 2 standard errors and about 99.7% of the sample means will be within 3 standard errors of the population mean.  Just as z-scores can be used to understand the probability of obtaining a raw value given the mean and standard deviation, we can do the same thing with sample means.

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