Strict Average versus Power-law

We would logically assume that students proficiency in a particular skill will improve over time. Thus, as their skill improves their score on assessments for that skill should increase.

The strict average weights all attempts equally, so that the first attempt is just as important as the most recent attempt; even though the student may have made great improvements in that particular area.

The power-law operates under the assumption that as students make improvements in the skill area, their summary score for that skill should improve and follow their assessment scores as they improve.

Consider a student who, on 4 assessments on the same skill scored 0, 2, 3, and 3.5 respectively. As we would expect, their score increased over time. The strict average for their 4 scores is 2.125. The question is: Does 2.125 summarize their current proficiency? It most certainly does not, when one takes into consideration their continuing improvement and their most recent score.

The power law, on the other hand, follows their improving scores and leads to a summary grade of 3.5.

On the graph below you can see the student's scores as blue boxes, the power-law curve in red, and the average in green.

Power law versus strict average

Click Here for a graphical, interactive demonstration of power-law versus average.

The beautiful part about the power-law is that it not only summarizes the students current proficiency level, it also mathematically can be followed backward to show how the students proficiency has changed over time.

Why not just use one summative score as the student's proficiency level?

Two reasons: students don't always perform to the best of their abilities every day and in every measurement there is some amount of uncertainty. Tracking progress over time helps to smooth out the effects of both of these variables.

What if the student's scores aren't so predictable?

As I just mentioned, there are things that make scores vary. There is a certain amount of uncertainty in every measurement. We can assume that every assessment score is accurate within a certain amount of confidence. Thus, if we take the scores over time we can follow the trend and find the closest fitting power-law to summarize the student's proficiency.

The graph below shows another student who scored 1, 3, 3, and 2 on a series of assessments on the same skill. The student's strict average (shown in green) comes out to 2.25. The power-law, in blue, follows an exponential curve that most closely fits the student's scores, showing that over time the student's scores are improving and that we have mathematical evidence to report their proficiency in this skill as 2.75 (the red dot on the power-law curve). It also gives us a prediction that if the student were to attempt the skill again, we would expect them to score around 3.

Power law vs Strict average #2

What if a student's score decreases over time?

This does happen, unfortunately. In those cases the power-law follows their grade downward and gives them a less-than favorable summary. In these instances, the strict-average gives the student a better overall score. gives you the option to summarize each individual standard by the Power-law, the strict average, or to choose which method gives the student the best outcome.

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