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When a teacher gives an exam in class, how does he or she decide if the test scores were good or bad? This lesson focuses on classroom assessment, including specifically how to analyze the variability of scores within a given group of students. We'll discuss both standard deviation (how much the scores vary from each other) and how to interpret bell curves (a graphical representation of the distribution of scores).
Imagine you're a teacher and you give your students a test over their understanding of the United States Revolutionary War. Once you grade the test, how do you know what the scores mean in terms of how well the students did? Another lesson talks about how you can quickly summarize the pattern of scores, or the average, using the concepts of mean, median, or mode. However, these concepts relate to summarizing how high or low the scores were in general. In addition to a quick average summary, you might also be interested in how much variability there was in the students. In other words, did all of the students get similar scores to each other? Or did some students do really well while other students in the same class did really badly?
The purpose of this lesson is to talk about how you can learn about the variability of scores in a classroom environment and why that might matter. We're going to cover two important concepts, which are standard deviation of scores and how to construct and interpret a bell curve based on student scores.
Let's start by using an example. Imagine you teach a class with 20 students, and they take a test with 20 multiple choice questions about the Revolutionary War. Imagine that the grades you get back from scoring their tests look like this:
Student #1: 20 | Student #11: 10 |
Student #2: 17 | Student #12: 10 |
Student #3: 16 | Student #13: 10 |
Student #4: 14 | Student #14: 8 |
Student #5: 14 | Student #15: 8 |
Student #6: 12 | Student #16: 8 |
Student #7: 12 | Student #17: 6 |
Student #8: 12 | Student #18: 6 |
Student #9: 10 | Student #19: 4 |
Student #10: 10 | Student #20: 3 |
Now you want to know the basic variability within the classroom. So, did the students' scores kind of clump up all together, meaning the students all showed about the same amount of knowledge? Or did the scores vary widely from each other, meaning some students did great whereas other students failed the test?
The answer to this question can come very precisely by calculating a standard deviation. A standard deviation is a number that indicates how much a group of scores vary from one another on average. Another way to say this is that the standard deviation tells you the standard, or typical, amount that scores in a group deviate, or change, within that group.
So let's look at our example from the Revolutionary War test. It looks like the scores have a pretty big range; the top score was 20, while the bottom score was only 3. But the standard deviation will give you an exact number telling you how much they really do vary within the group.
To calculate the standard deviation, it's a little complicated. First, you need to know the mean, or average, score. This equation is covered in another lesson, so we'll skip that equation for now. Just believe me that the mean score out of this group is 10.5, which is right in the middle of the pack.
The next step is that you take each score and subtract the mean from it to get a difference. So if you look at the screen, you can see that 10.5 has been subtracted from each score. For example, the top score was 20, and we subtract 10.5 from that to get a difference of 9.5. For the bottom half of the students, you end up with a negative result. For example, 3 - 10.5 equals negative 7.5. For standard deviation, it doesn't matter if the difference is positive or negative, so you can just ignore the negative sign and keep the actual number of 7.5. Do that for all of the scores in the group.
Now you just take all of those difference scores, add them up, and divide by the number of scores you had. The number you get will be the average amount the scores were different from the mean of 10.5. Here, when we add up all the difference scores, we get a total of 66. Now, we divide 66 by 20 (the number of students), and we get a final score of 3.3. The standard deviation of scores is 3.3, meaning that the average amount scores vary from the middle is between 3 and 4 points on the test.
Now that we have our standard deviation of 3.3, what the heck does that mean? Well, it just gives us an idea of how much the scores on the test clumped together. To understand this better, look at the two distributions of scores on the screen. The one on the left shows scores that are all very similar to each other. So, because the scores are all close together, the standard deviation is going to be very small. But, the one on the right shows scores that are all pretty different from each other (lots of high scores on the test, but also lots of failing grades on the test). For this distribution, we'd have a high number for our standard deviation.
So why do we care about standard deviation at all? Well, a teacher would want to know this information because it might change how he or she teaches the material or how he or she constructs the test. Let's say that there's a small standard deviation because all of the scores clustered together right around the top, meaning almost all of the students got an A on the test. That would mean that the students all demonstrated mastery of the material. Or, it could mean that the test was just too easy! You could also get a small standard deviation if all of the scores clumped together on the other end, meaning most of the students failed the test. Again, this could be because the teacher did a bad job explaining the material or it could mean that the test is too difficult.
Most teachers want to get a relatively large standard deviation because it means that the scores on the test varied from each other. This would indicate that a few students did really well, a few students failed, and a lot of the students were somewhere in the middle. When you have a large standard deviation, it usually means that the students got all the different possible grades (like As, Bs, Cs, Ds, and Fs). So, the teacher can know that he or she taught the material correctly (because at least some of the students got an A) and the test was neither too difficult nor too easy.
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So, we can get a good idea of the pattern of variability using this idea of standard deviation. However, there's one more way you can look at the pattern of scores. That's our last topic for this lesson, and it's the idea of bell curves.
To construct a bell curve, you plot the test scores in a graph. The x-axis is for the score received and the y-axis is for the number of students who got that score. So still using our same example of 20 students who took a test with 20 questions, you can see here the pattern that shows up on the graph.
There's a big bump in the middle, showing the five students who got the middle score of 10. Then, the graph tapers off on each side, indicating that fewer students got very high or very low scores.
Remember that before, we said that most teachers will want their students' scores to look kind of like what we see here. We had a lot of scores that fell in the middle (indicated by the big bump), which might be like a letter grade of a C. We had a few students who did really well (which might be like the grade of A) and a few students who did badly (in other words, they got an F). When you have a bell curve that looks like this one, with a bump in the middle and little ends on each side, it's called a normal distribution. A normal distribution has this bell shape and is called normal because it's the most common distribution that most teachers would see in a classroom.
Now, imagine that most of the students got an A on the test. What would that bell curve look like? It would look something like this one. You can see here that the bump falls along the right side of the graph (where the higher scores are), with it tapering off only on the left side, showing that most students got high scores and only a few got low scores. The exact opposite would be true if most students got an F, which would look like this graph. When a bell curve is not normal and is instead weighted heavily on either side like this, it's called a skewed distribution. Skewed distributions are both less common in classrooms and usually less desirable.
In summary, a standard deviation is a number that indicates how much a group of scores vary from each other, on average. Standard deviations are important because they tell you how much a group of students varied on any given test. They might be able to tell you if the test is too easy or too difficult.
You can graph the distribution of any group of scores using a special graph called a bell curve. A normal distribution has most scores falling in the middle of the range, whereas a skewed distribution has most scores falling on one extreme end or the other. Think about the typical scores that you used to get in middle school or high school. Where would your grades fall on a bell curve?
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