In my math class we have a measure that we call the Bandini Line. This is the score that students must surpass to achieve a passing grade on their assignments. The name comes from a fertilizer commercial that aired in Southern California in the mid to late 1970's in which a skiier stood atop a gigantic pile of cow manure (Bandini Mountain) and unsuccessfully attempted to ski down. He face planted in the manure and that was the humor in the commercial. It was pretty low brow, but to a Junior High School boy it was the funniest thing ever.
In my math class the goal is to avoid the skiier's fate by remaining above the Bandini Line, which is 80 percent.
With the Bandini Line at 80 percent, it means that the average score on a quiz is typically in the range of 84-87 percent. Some students do not reach Bandini and some students surpass the quiz total by doing bonus questions which award them extra points. Now think about that for just a minute. In order to be "average" in my math class the students need to 84 percent on math assignments that are, at a minimum, one grade level ahead and usually 2 or 3 grade levels ahead. That's really hard, and quite an academic achievement. That means that the "average" student in my math class performs above the 90th percentile nationally. This isn't to boast, it is simply to point out the imprecision of the notion of average.
The reason that I bring this up is that earlier this school year, standardized testing results were sent to parents that included a bit of data that compared their child's score to their peers in our community. Some parents where quite distressed when they read that their child scored in the 12th percentile in a particular academic area when their teachers had been reporting all year that they had been doing fine. How is this possible?
It is possible when a lot of kids do very well on a test and their scores are very tightly grouped. If the mean score on a test is 84 percent and the standard deviation is 3 percent, that means that 2/3 of the students scored between 81 and 87 percent. It means that 96 percent of scores fall between 78 and 90 percent; so a student can fall in the 75th percentile nationally and in the 4th percentile locally.
It means that average is a very nebulous thing to define.
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