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Those who aren’t familiar with the intricacies and difficulties of systemic measurement might not be familiar with the basic routines used to determine accuracy. The standard error is regularly used to determine the accuracy of a mean when compared to a larger sample size, and below, we’re going to show you how to calculate standard error when using specific formula. We’ll also include links to further literature about this common statistical exercise so that when the time arises that you need to put it to use, you’ll have plenty of knowledge at hand to do so.
Whether you’re a statistics nerd or someone looking for only the most basic understanding of standard error, read on to learn more!
Calculating a mean or median from a sample isn’t a particularly hard process. Depending on the nature of the study, it’s usually going to mean selecting a specific sample size relative to the larger population that you’re drawing from. After doing so, the next step in the process is to find that mean or median (or “average,” for sake of our concise explanation) from among that sample. Using that average, you can make some determinations about the larger population that you’ve selected the sample from.
However, as anyone familiar with mathematical statistics is going to tell you, such a simple process might not provide the most accurate of studies. After all, how can you be sure that the sample you’ve selected properly represents the larger population? How do you account for other, similar studies that have taken place from the same population, but with differing samples? These are all things that need to be taken into account, and in the world of statistics, that means a process is necessary for measuring that accuracy.
The formulae devised for standard error are that response to much-needed accuracy. And though they might seem fairly complicated–and are capable of becoming quite complicated–understanding what they’re used for isn’t particularly difficult. And because a sample size is always going to differ from the size of the population it’s representing, the standard error calculation is necessary. after all, if there were no difference between sample size and population, there wouldn’t be any need for a sample, right?
Below, we’re going to further introduce you to methods of calculating standard deviations between sample and population sizes. And as promised, we’ll be concluding this brief introduction to these formulae with further reading to invest in, if you’re interested in the study of statistics.
Mean, Median, Sample & Population
Any statistical study that’s meant to represent a larger population is going to require a sample. And as long as the sample is properly chosen, the mean or median of that sample can be applied to the population with varying degrees of accuracy. However, “varying degrees” can differ quite a lot. Thus, deviations between sample and population are almost as important as the information gleaned from approximating a mean or median.
“There’s no such thing as median income; there’s a curve, and it really matters what side of the curve you’re on. There’s no such thing as the middle class. It’s absoltutely vanishing.” – Marc Andreessen
Without a method of determining deviations, the information acquired from studying the sample becomes less relevant and applicable. In fact, it’s virtually useless to try to determine standard error unless you have the necessary information for determining standard deviations. The two are intricately connected, and even a surface-level examination of the two formula that we’re going to discuss below reveals this–if you don’t know the parameters of the represented population samples, what use is the standard error of those samples going to be?
The Importance of Deviations
Deviations are going to change depending on the scenario that they’re being measured in, but there are a few things that will almost always be applicable when determining them. When looking at a set of data from a determined sampling from a larger population, examine the data points in relation to the established mean. If they’re spread far above and below the mean (or average), you’re looking at a greater deviation–which usually represents a more volatile study, within a specific sample. Most studies that expect any sort of accurate result have relatively low standard deviation and standard error–this represents most data points being clustered close to the mean. Studies that have data points spread evenly across a wide range of variables have less chance of being accurate and usually helps researchers to know that they need a new angle of inquiry.
If the data points are relatively closer to the mean or average, the deviation values will be smaller, too. This will be important for when you’re calculating the standard error from a set of data points. The measurement of this deviation in a given set of data is usually referred to as the standard deviation.
Standard Error & Standard Deviation
Though they sound similar, standard error and standard deviation are actually two different things, even though they’re both used to determine the amount by which data is spread across sample sizes.
Standard error utilizes the data from samples to determine the spread of that data. And standard deviation plays a critical part in the calculation of that since it’s used to determine the parameters of the population that are being represented. One fits into the other, and thus, the two are inseparable; if you want to calculate standard error, you’re going to need to calculate standard deviation if you want your results to be accurate. Without the parameters, you’re not going to be able to calculate the standard error of a sample with any degree of accuracy.
Calculating Standard Error
As long as you have the necessary information to do so, there are formulas that can be used to calculate the standard error in situations like these. The standard error of the mean (SEM) represents the standard deviation of the sample mean estimate taken from an overall, larger population mean. As we’ve described above, it’s a method that you’re going to use to determine the accuracy of a sample mean, alongside its relation to the population that the sample was taken from.
In the above formula, there are several variables that you’ll need to know if you want to complete the equation. Defined below, each of them is essential:
- SEM: The standard error of the mean
- s: The standard deviation of the sample (see below)
- n: The number of observations of the sample
To calculate the standard deviation of the sample, you’re going to need more extensive information from each of the sample sets of data that you’re using. These must all be taken from the larger population size. Your calculations of standard deviation and standard error will be far more accurate if all samples are the same size.
As you can see, there are several more necessary variables for finding the standard deviation of the sample. Unless you’ve already defined them all, each of these is going to require some further calculation. They’re all defined below:
- s: The standard deviation of the sample
- x1-xn: The separate sample data sets that you’re calculating, to determine the standard deviation of the sample
- x̄: The mean value taken from a sample data set
- N: The size of the sample data set, taken from the population
Using these formulae, you’ll be able to successfully determine the standard error or your sample size, in relation to the population. It’s a fairly standard exercise in statistical science and general research, whenever there’s a disparity between several different samples and the population they’re meant to represent. Of course, we’ve explained the process as concisely as we absolutely could. There are quite a few different circumstances where the formulae can become more complex. For example, determining the parameters in the standard deviation can become incredibly different when you’re working with a lot of different samples; the more samples that you’re working with, the more complex your formula is going to be.
However, the above formula does allow for a variety of different samples to be input, regardless of size or complexity. If you want further reading on the topic, refer to the links that we provided at the top of this section.
One might argue that these types of formulae and calculations are only useful for those committed to intricate statistical knowledge, but you’d be surprised by how versatile such data can be. Research scientists and even journalists interested in measuring rising and sinking trends can make use of standard error and standard deviation. It can reveal interesting things about populations and the samples derived from them; thus, if your profession regularly works with samples and populations, it can be incredibly helpful to keep these formulae close at hand.
Using the above steps, you’ll be able to figure out how to calculate the standard error of your study’s particular samples. Not only that, but you’ll be able to calculate the standard deviation necessary for the former formula. If you have any questions about either of these, feel free to hit us up in the comments section of this article! Otherwise, consider clicking through some of the verified links that we’ve provided, for further reading.