Standard Error

Photo by Sarah Kilian

So, What Are Standard Errors?

So standard errors are basically like standard deviations but not really. A standard deviation as we may know is a measure of spread around a mean of the random variable we are analyzing. A standard error on the other hand can be thought of as a measure of spread around a sampling distribution mean. In other words, if we sample many times from a population and analyze these means, we can essentially compute a standard error which is interpreted like the standard deviation of that sampling distribution.

What are Standard Errors Used For?

Standard errors are important for the following:

1) Estimating confidence intervals of any statistical parameter

2) Conducting hypothesis test of any statistical parameter

3) Like we just mentioned, measuring the spread of a sampling distribution

4) Statistical simulation and of course conducting a sampling distribution

5) Sample size calculation (When conducting research studies)

So, the bottom line is that standard errors are an inevitable part of statistical analysis.

Types of Standard Errors

So, guys to keep things short and sweet, we will document a handful of commonly used standard errors, but don’t sweat the details of these formulas, the main idea is that the interpretation I show you guys at the end is the same.

The standard error of a mean (sample mean)

s/√n=s²/n

The standard error of a single proportion

√((p ̂(1-p ̂))/n)

The standard error of a regression parameter estimate (MSE stands for Mean Square Error)

√(MSE/(∑▒〖(x_i-x ̅)〗² ))

We will cover standard errors for the more commonly done statistical tests (including the one on the regression parameters) in other articles and videos. The key take-away is that the standard error depends on what parameter is being estimated and remember it is an estimate that represents the sampling distribution of that parameter.

Overall standard errors help us understand how estimates of a statistic can vary given repeated sampling. It is a fundamental metric in frequentist statistics that is probably used and often mis-used.

One key caution when reporting or analyzing data with standard errors is to never report them with a mean, unless that mean is a simulated mean or a mean from a sampling distribution. You should stick with reporting the standard deviation for a single sample mean. (Why, because I said so, just kidding. It is actually incorrect to report the standard error (SE) because they are meant for assessing spread around a bunch of means or point estimates not a single mean).

Thanks for reading. If you guys liked this article be sure to check out my channel, and like and subscribe. Until next time. Happy mining.

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