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Average, Middle, and Most Common: Stats Every Teacher Should Know
Educational Psychology · Chapter 8: Assessment, Testing, and Making Sense of ResultsSection 8.8 · lesson 77 of 1009 min · free with a free account
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Next lesson: Standardized Tests: The Good, the Bad, and the Complicated →← Compared to Others vs. Compared to a Standard: Norm vs. CriterionNext lesson: Standardized Tests: The Good, the Bad, and the Complicated →
In this lesson
Center, spread, and how a single number can hide an entire class.
- Distinguish mean, median and mode
- Explain range, variance and standard deviation
- Read skew from the mean and the median
- Turn class statistics into an instructional decision
Transcript
Everything said in the video, section by section (1,230 words). Read along, or search it.
Assessment, Testing, and Making Sense of Results
Average, Middle, and Most Common
A class average of eighty-two percent sounds like a clear result. It could describe a class where everyone scored around eighty-two. Or a class where half scored ninety-five and half scored sixty-nine. Those two classrooms need completely different lessons tomorrow. This section is about the statistics that tell them apart. First, measures of central tendency. Mean, median and mode. Then measures of variability. Range, variance and standard deviation. Then how to read skew by comparing two of those numbers. And a five-step routine for after every major assessment. None of the math here is difficult. What is difficult is remembering that one number is never enough.
Central Tendency
Three Ways to Say the Middle
Three statistics describe where the center of a distribution lies. The mean, the median, and the mode. The mean is the arithmetic average. Add all the scores, then divide by the number of scores. It is the most mathematically tractable of the three. And it is the one most sensitive to extreme outliers. The median is the middle score, once all scores are ordered. It is a positional measure rather than a mathematical one. Which is precisely why it resists outliers. The mode is the most frequently occurring score. It is the only one of the three meaningful for nominal data. That is, for categorical data with no natural order.
Outliers
What One Score Can Do to a Mean
Here is exactly how much damage one outlier can do. Take a class of twenty-five students. They all score between seventy and eighty-five on a test. All except one student, who scores twelve. That single score pulls the mean substantially downward. And the resulting mean misrepresents the typical performance badly. Now look at what the median does with the same data. It would remain in the seventies, even with the twelve included. A median is a position, and one outlier cannot move a position far. So the reporting risk is real, and it runs deeper than reporting. Teachers who report only the mean may distort their own understanding. Not just what they tell others about class achievement.
Bimodal
When Two Peaks Mean Two Classes
One case deserves particular attention. The bimodal distribution. That is a distribution with two distinct peaks. When you find one, both modes should be reported. Because two peaks may indicate two distinct subgroups within the class. That is an instructional finding, not a statistical curiosity. And consider what the average does in that situation. A single average sits somewhere between the two peaks. It describes a student who may not exist in the room at all. Everyone is either well above that number or well below it. So the mean is not merely imprecise here. It is describing a person nobody in the class resembles.
Variability
Center Is Only Half the Story
Central tendency tells us where scores cluster. Variability tells us how spread out they are. The simplest measure is the range. That is the difference between the highest and lowest scores. A class with scores from fifty-five to ninety-eight has a range of forty-three. It is easy to calculate, and it has two real weaknesses. It is heavily influenced by a single extreme score. And it ignores the distribution of everything in between. Variance is the average squared deviation from the mean. Mathematically convenient, but expressed in squared units. Which makes it difficult to interpret directly. Standard deviation is the square root of the variance. That returns it to the original score units, which is why it is used.
Reading The SD
One Number, Two Different Classrooms
A standard deviation is not only a descriptive number. It is an instructional signal, and it should be read as one. A large standard deviation relative to the mean indicates heterogeneity. That is a wide range of achievement levels in one room. It calls for differentiated instruction. Flexible grouping, tiered assignments, and differentiated support. A small standard deviation indicates relative homogeneity. Most students are performing similarly to one another. There, the teacher can focus on moving the class forward together. Notice what that means about the order of operations. The standard deviation tells you the shape of instruction needed. Before you have looked at a single individual student's score.
Center And Spread
Why the Mean Alone Misleads
Which brings us back to the example from the opening. Reporting only the class mean gives an incomplete picture. And a potentially misleading one. A mean of eighty-two percent could be a uniform class. Everyone scoring somewhere near eighty-two. Or it could be a bimodal class. Half scoring ninety-five percent, and half scoring sixty-nine. Those two classrooms are nothing alike. So always pair the mean with the standard deviation. And inspect the full distribution before drawing conclusions. The mean cannot tell those two classes apart. And the mean is the number most teachers report.
Skew
Comparing the Mean to the Median
There is a fast way to detect the shape of a distribution. The mean and the median, together, tell a story. If the mean is greater than the median, the distribution is positively skewed. A few very high scores pulled the mean upward. Which means most students actually scored below the mean. If the mean is less than the median, it is negatively skewed. There, a few low outliers pulled the mean down. And a roughly equal mean and median suggests approximate normality. Two numbers, one comparison, and you know the shape. Then plot a simple frequency distribution to confirm it. Tally how many students scored in each ten-point band. That makes the pattern visible, and reveals clusters needing support.
The Six
Each Statistic, and When to Use It
Six statistics, with the condition under which each is the right choice. The mean is the sum divided by N. Use it when distributions are normal and symmetric. The median is the middle score in an ordered set. Use it when outliers are present, or the data are skewed. The mode is the most frequent score. Use it with categorical data, or bimodal distributions. The range is the maximum minus the minimum. Use it for a quick overview, remembering it is sensitive to outliers. Variance is the average squared deviation from the mean. It is the foundation for standard deviation, not directly interpretable itself. And standard deviation, the square root of variance, is the primary measure of spread.
Classroom Example
Six Students, Not Thirty
Here is a teacher turning four numbers into a decision. After a social studies unit test, Ms. Park calculates her statistics. Mean of seventy-six. Median of eighty-one. Standard deviation of fourteen, and a range of fifty-two. The median is above the mean, which signals negative skew. That means a few students with very low scores. So she plots a frequency chart to see where they are. And she discovers a cluster of six students scoring below sixty. Now notice the decision she does not make. She does not assume the whole class needs reteaching. She targets a small-group reteach session for those six students. Preserving instructional time for everybody else in the room.
How To
A Five-Step Routine After Every Test
To finish, a five-step routine for after any major assessment. First, calculate mean, median and standard deviation in a spreadsheet. Second, compare the mean and the median to detect skew. Third, plot a bar chart by ten-point score bands. Fourth, identify clusters, either for reteaching or for enrichment. Fifth, review individual items failed by more than forty percent of students. Review those for possible reteaching. That fifth step is doing something different from the first four. Steps one through four analyze the students. Step five analyzes the test itself. And sometimes the honest answer is that one question was badly written. Teachers who know their class statistics make more targeted decisions. More targeted than teachers relying solely on intuition.
About Educational Psychology at UPI Study
Educational Psychology (Psychology 120) is a self-paced online course of 100 video lessons in 11 chapters, from memory, conditioning and child development to motivation, individual differences, assessment, instruction and educational technology. It carries a NCCRS credit recommendation for 3 semester credits.
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