BIO - MATH: Standard Error of the Mean [LESSON]
MATH: Standard Error of the Mean
IMPORTANT NOTE: For this lesson, you will need a calculator AND your AP Biology equation sheet.
Click here to download the Equation Sheet. Links to an external site. *It would be a good idea to print this sheet for quick reference*
For the expected math knowledge coming into the course, see this handout “Introduction to AP Biology data analysis” Links to an external site..
Welcome to our first MATH lesson! Since science is really just applied math, it is interwoven throughout the entire course. For this first unit, we will learn our first statistical concept which is the standard error of the mean. Go ahead and locate this equation on your equation sheet. You will also see that the equation for standard deviation is located on the AP Biology equation sheet, but the AP exam will likely not ask you to hand calculate this. Please note that the equation sheet gives you a LOT of information about what each variable in the data set means.
The standard deviation and standard error calculations are used when the data set approximates a normal distribution, or “bell curve” as in the image below.
Remember that the y-axis is frequency, so the image above displays the frequency of individual observations in a data set, and the closer you are to the mean, the more frequently you see that observation. Normal distributions have the following features:
-symmetric bell shape
-mean and median are equal; both are located at the center of the distribution
- approximately 68% of the data falls within ±1 standard deviation of the mean
- approximately 95% of the data falls within ±2 standard deviations of the mean
- approximately 99% of the data falls within ±3 standard deviations of the mean
Standard Deviation
This is a measure of how spread out the data is. The larger the standard deviation, the more spread out the data. We can draw conclusions in science more easily when the data has a smaller standard deviation. This is because it becomes more apparent that this trend in the dependent variable happens frequently and we can attribute it to the change in the independent variable. In the image below, we can see that the two data sets have the same mean but the blue data set has a much higher standard deviation. The red data set has a much lower standard deviation and so the data is less spread out and is instead more clustered around the mean. Check out your equation sheet for a refresher on calculating standard deviation.
Standard Error of the Mean
The standard error of the mean (SEM) estimates how well the sample mean (¯x) estimates the true mean (μ) of a population. It is similar to thinking about where our sample mean might fall in a normal distribution of all the sample means we may have collected from a population. The equation for standard error is
SEx=s√n, where s is the standard deviation of the data set and n is the sample size collected. Please watch the video below carefully and take notes about standard error.
Watch the Standard Error of the Mean video below.
Use the practice below to calculate the standard error of each data set.
As you saw in the video, we often use the standard error to display error bars on a bar or line graph. The value of the upper bound of the error bar is two standard errors above the mean and the value of the lower bound of the error bar is two standard errors below the mean. Together, these values encompass a range where there is a 95% probability (confidence interval or CI) that the true mean (μ) may lie. Thus, if the error bars on a graph do NOT overlap on the y-value, we can say that there may be a significant difference between the two means. However, if the error bars overlap on the y-value, we cannot say that there is a significant difference between the two.
Use the tools to practice both calculating error bars and interpreting error bars on a graph in the Errors Bars Self-Assessment below. Select Next to move through each question.
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