Measures of Variability

Updated: Nov 10

Video Transcript

Thanks for tuning into this video on measures of variability. My name is Matthew Courtney. I am an education researcher and data consultant specializing in continuous improvement in schools. In this video, I am going to discuss measures of statistical variability and how you can use them for school level decision making.

Measures of variability are standardized metrics that help you understand how spread out your scores are. Generally, in education we want to see low measures of variability. That means that all of our students are performing around roughly the same level. However, it is much more common to see high measures of variability among our students, with groups of students scoring above or below their peers. By calculating measures of variability, we can better identify students who need help or classes that are falling behind.

There are three measures of variability that we will discuss in this video: range, interquartile range, and standard deviation.

The range is the easiest one. It is simply the distance between the highest score and the lowest score in your distribution.

Interquartile range is related to the range, except it is the distance between the first and the third quartile. Quartiles divide your scores into four even chunks. Starting with the minimum score, the first quartile divides the bottom half of scores into two. The median is the exact middle score. The third quartile divides the upper half of scores into two chunks. This measure shows you how far apart the middle half of your students are performing and allows you to control of outliers.

Here is a simple example using a distribution from one to nine. One is the minimum score and nine is the maximum score. Five is the median score, or the exact middle. The first quartile is three – it is the number exactly between the minimum and the median. The third quartile is seven, the number exactly between the median and the maximum.

Finally, let’s look at standard deviation. Of the three measures of variability we will discuss, this is perhaps the most commonly reported in education performance reports. Formally, it is the square root of the variance of a distribution. It helps you understand where your students scored in relation to the mean. A low standard deviation says that most students performed close to the mean while a higher standard deviation tells you that your scores are more spread out.

This graph is the easiest way to understand what is going on with standard deviation. Generally, student performance is going to show up on a normal bell-curve distribution. When it does, the standard deviation works like this. The middle line, marked here with a zero, is the mean, or average. Each vertical line on the graph represents a standard deviation. Typically, 34.1 percent of students will score on either side of the mean, with 13.6% of students being one standard deviation away, 2.1% of students being two standard deviations away, etc.

Let’s look at an example with numbers. Let’s say you give a test on a 100-point scale. Your students scored an average score of 65 with a standard deviation of 15. Your mean is 65, so it goes right in the middle. 68.2 percent of students scored between 50 and 80. That is within one standard deviation. The rest of your students scored beyond that mark.

Let’s take a look at standard deviation in action.

Principal Robinson has been given funding to hire an additional interventionist to work with students in second grade. She has three second grade classes who must share the time of the interventionist. She wants to focus on an equitable distribution of time, ensuring that the students who need the additional services have access to it - regardless of which homeroom class they are in. She uses measures of variability to from the back-to-school level assessment to create the interventionist schedule.

Here is Principal Robinson’s data. We see the mean, range, interquartile range, and standard deviation for each of the three second grade homeroom classes. How should the interventionists time be divided?

Let’s start with Homeroom Two. Homeroom Two is clearly outperforming the other two classes. They have the highest average and the majority of students have scored between 83 and 85 on the assessment. This class should get the least amount of targeted time with the interventionist.

Homeroom Three has the lowest average score of all three, so at first glance you may think that this course needs the most targeted intervention time. However, the standard deviation and interquartile range are low – meaning that MOST students scored around that 52 percent mark. This course would be better served by some regular visits from the instructional coach who can help this teacher improve their whole group teaching skills.

Homeroom One should get the majority of the interventionists time. Most students in this class scored between 57 and 67 percent on the assessment. The scores are still low, but the variation is high, with some students scoring as high as 73 and as low as 52. Every classroom probably has one or two kids who need intervention, but there are some students who can easily be targeted for improvement in this classroom. Principal Robinson should start the year with the interventionist spending more time here than in the other two rooms.

As with any decision making process, Principal Robinson should continue to monitor this decision and adjust the interventionist schedule as necessary throughout the year.

Now, take some time to practice applying these skills to your classroom! Access the data from a recent assessment and divide it among the various sections that you teach. Use the tools in your spreadsheet software, or the free tools available at, to answer the following questions: Which class has the highest performance? Which class has the greatest variation? Which classes would benefit from whole group re-teaching? Which classes would benefit from targeted, individualized, intervention?

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