Welcome to #BeyondTheMean! Check out this post to see what this blog is all about.
Educators have access to a lot of data. Unfortunately, a list of test scores doesn’t do you much good when you are trying to understand a continuous improvement problem or make decision about a classroom, school, or system. In order to make use of that data, education decision makers must summarize and describe it using descriptive statistics. In this post, I want to discuss the six descriptive statistics used in education and talk a little about how you might use them to inform your decision making.
Mean
The mean is the statistical average of a range of scores. You calculate the mean by adding up all the individual scores and then dividing the sum by total number of all scores. It is a figure that helps you understand where your students performed as a group. The mean, along with the median and mode, is a measure of central tendency, meaning that it summarizes a range of scores around its most central point.
The mean is a valuable statistic to calculate and is often the first statistic calculated when summarizing your data. It is easily understood by most people and easily compared to the mean of previous occurrences. It is also a pre-requisite to many more rigorous statistical calculations deployed by analysis seeking to make inferences about a set of data.
One downside to the mean is that it is heavily swayed by outliers. If you have a few students who scored far above or far below the rest of the students, the mean will be skewed in that direction. Remember teasing the “curve busters” in your classes in high school? Those students “broke the curve” because they scored so far above the rest of us that the average on the test was higher than it should have been – discouraging our teachers from granting bonus points across the board. This is why I always say that we must look #beyondthemean when analyzing our data.
Median
One way that we can look #beyondthemean, is by examining the median. The median is the score in the exact middle when you line them all up. If you have an even number of scores, then the median is the average of the two scores in the middle. You can only calculate the median if you have ordinal data – that is data that can be placed in a logical order, such as a test score.
In a perfectly distributed set of scores, the mean and median should be the same because they will both be in the exact middle of the distribution. Unfortunately, a perfectly distributed set of scores is a freak of nature that almost never happens. As such, the relationship between the mean and median is important because it can give you a feel for the skew of your data.
The skew describes which way your data leans. If the median is higher than the mean, then the data is said to “skew to the left” or to have a negative skew. That means that the outliers in your data are on the lower end of the spectrum. When your median is lower than the mean, the opposite is true. This data “skews to the right”, or has a positive skew, meaning that the outliers in your data are on the upper end of the spectrum.
Mode
The mode is simply the number, or numbers, that occur most frequently in a distribution of scores. The mode is best applied to categorical data and can be used to easily explain which category has the highest number of inputs. A distribution of scores may be unimodal, containing only one mode, bi-modal, containing two modes, or mulit-modal, containing more than two modes. By examining the mode, you can quickly spot clusters in your data that may contain useful information.
Range
The range is the difference between the highest and lowest scores in your distribution. Along with the standard deviation and quartiles, it is considered a measure of dispersion. Measures of dispersion help you to see how spread out your scores are – or how side the gap is between your highest performing student and your lowest performing student.
This is valuable information for educators working to ensure that all students achieve at high levels. The range is a quick calculation that can show you if your students are performing together as a group or if you have some students performing way above or way behind the rest of the pack. Generally speaking, when looking at educational data, we want to see low ranges. That indicates that our students are moving together as a group.
Standard Deviation
The standard deviation is a more rigorous measure of dispersion as it applies some heavy math to the situation. A low standard deviation indicates that most of the scores are clustered around the mean while a high standard deviation indicates that the scores are more spread out. A standard deviation of zero, for example, would mean that all students scored the exact same thing.
Standard deviations help us to group students around the mean. In a perfect distribution, 68.2 percent of your students will be within one standard deviation of the mean, 34.1 percent above the mean and 34.1 percent below the mean. As we get farther away from the mean, we should see fewer students falling into the various standard deviation categories. For example, 13.6 percent of students should fall within two standard deviations of the mean, 2.1 percent should fall within three standard deviations, and 0.1 percent should fall within four standard deviations.
So, what does all of that mean? Let’s say that we have a set of student scores with an average of 100 and a standard deviation of 10. We can assume that 68.2 percent of our students scored within 90 and 110 points, that is, 10 points on either side of the mean. We can assume that 13.6 percent of our students scored within two standard deviations, meaning they scored either between 80 and 90 points on the low end or 110 and 120 points on the high end. We can continue in this manner adding or subtracting in clusters of 10, our standard deviation.
Quartile
The final descriptive statistic we will discuss is the quartile. The quartile chunks your data into four even parts. In an evenly distributed set of scores from 0 to 100, the quartiles would be 0-25, 26-50, 51-75, 76-100 These quartiles help us to group students and see where clusters of students fall on the spectrum. Of course, no distribution of scores will ever be perfectly even from 0 to 100, so let’s consider a more real world example.
Imagine that you gave a test with a minimum score of 62 and a maximum score of 98. The quartiles for this may be 62-69, 70-79, 80-90, and 90-98. This tells us that a quarter of our students scored between 62 and 60, another quarter between 70 and 79, another quarter between 80 and 90, and another quarter between 90 and 98. You can see that they aren’t evenly distributed, likely because some students scored the same. With this information in hand, we can make better informed plans as we seek to improve teaching and learning in our building.
In this short post, I have sought to introduce you to the six most common descriptive statistics found in education. Hopefully you will be able to interpret them correctly the next time you run across them in educational research. If you want to get a jump start on your data analysis, check out the free auto-analysis tools housed in The Repository. I have built six tools for your use. Simply upload your data and let the computer do the work for you! Good luck on your journey friends, and let me know how I can help.