Have you ever had a situation where one grade destroyed your average? Wouldn't you like a way of proving that your work was actually pretty good with that one exception? The standard error gives you such a chance.
Mean & Standard Error Definition
Let's say that you've just collected a great deal of data about something that can vary wildly, like the life spans of sea turtles. Some turtles barely make it out of the shell, others are caught in fishermen's nets from age five onward, and still others have been estimated to be more than 100 years old when they finally died. Clearly, you have a lot of variation in this data. Luckily, there are two ways that can be used to help make some sense of it all - the mean and the standard error. The mean of a sample is the average value of all the individuals in the sample. Meanwhile, the standard error shows how accurate your mean is by comparing it to the mean of all the values of an item that exists.
Finding the Mean
Chances are that you've been finding the mean or average of data sets for quite some time. However, let's review it just in case. To find the mean of a set of data, simply add all the values of the data together and divide by the total count of data points. At the end of every term, you get a grade that is an average of your performance in assignments throughout the period. For example, if your grade was solely based on the results of five tests, and you got a 98, 94, 79, 83, and 88 on those tests, their mean would be your grade. Added together, those numbers are 442. We divide that total by 5 as there were five tests. As a result, your final grade in that class for the quarter in question was an 88.
The Mean Isn't Perfect
The mean has some real uses, but also some real problems. In the example about your final grade, your performance had been that of a solid B or B+ student, hence the 88. However, the same mean could have been reached had you gotten 100s on the first four tests then completely failed the last one, getting a 42. For whatever reason, your performance tanked on that last assessment, and the mean says that your performance was the same as a student who had not done as well across the whole of the term. Wouldn't it be useful to know how accurate the average is across the data set? Luckily there's a way for that to be found.
Finding the Standard Error
A student with four 100s and a 42 is likely a different student from the one who gets a 98, 94, 79, 83, and 88. However, their means are identical. Something that they would be very different in is their standard error, the measure of how accurate a mean is in relation to the expected outcome of the real data. If that sounds like a bunch of jargon, think about it like this. Let's say that those five tests were actually not the sole grades for a course, but instead five assessments chosen at random. Five assessments chosen from a total of six assignments would be much more accurate than five assessments chosen from fifty. Standard error lets us quantify that difference. To find the standard error, take the standard deviation of the sample set, then divide it by the square root of the sample size.
Example of Finding the Standard Error
To see how this works, let's find the standard errors of the data sets above, assuming that each sample was taken from a collection of 25 assessments. In the first example, the standard deviation was 7.7653. Divide that by the square root of 25, which is 5, and we get a standard error of 1.553. Now let's go to the other set. The set of four 100s and a 42 had a standard deviation of 25.9383. That, divided by five, gives us a standard error of 5.188. As a result, we can assume that the mean for the first set of numbers is much closer to the reality of the whole set of 25 than the second set.
In this lesson, we examined the concepts of mean and standard error. The mean is simply the average of a sample, while the standard error is a measure of how accurately that mean reflects the mean of the total population.
Using the example of two sets of grades, we saw how identical means could have very different component grades and how standard errors helped to make some sense of this variation.
- To find the mean of a set of data, simply add all the values of the data together and divide by the total count of data points.
- To find the standard error, take the standard deviation of the sample set and then divide it by the square root of the sample size.
Vocabulary & Definitions
Mean: The mean is the average of a set sample that is found by adding each number from the sample and dividing it by the sample size.
Standard Error: Standard error is the error rate that measures how accurate the sample mean is to the population mean.
Working your way through the lesson will prepare you to:
- Define the mean and standard error
- Summarize the process of finding the mean and standard error