midnight Target: The flaw in your math miko, is that you assume the test was made easier.
I guess you've missed those statements of mine or have trouble with reading comprehension:
miko: The achievements of blacks and whites could have increased or the passing score could have became easier to achieve...
miko: I would not speculate whether there was real performance improvement of both whites and blacks on the test or the test was "dumbed down"
What "flaw in my math" are you talking about? I have calculated four numbers - four gaps between average scores of the groups based on their treshhold values using a basic statistical technique. Each case calculation is completely independent from the other three, does not use any data from the other three cases and does not depend on the existance of the other cases at all.
Whether the test was dumbed down or performance improved only has a bearing on why the treshhold values changed from year to year. Since it does not matter in the least why the treshhold numbers are what they are, only their numerical values relative to each other in that particulat case, how could it possibly affect the formula?
The only assumptions I made are:
1) that the test-scoring abilities are normally distributed (Gaussian bell curve) - which is not only a well know fact but is a requirement for creating and calibrating the valid tests.
and
2) that black standard deviation is about 0.9 of the white one - a minor assumption not substantially affecting the results and easily justifiable.
In a normal distribution it is a simple step to convert from a percentile to the standard deviations from the mean. That allows us to find the mean score - the score that half of the sample scored below and half above.
Knowing two scores for two populations, we calculate the gap - independently for each case. The gaps just happen to be very close in all four cases which shows that the relative performance of blacks compared to whites did not improve. Both blacks and whites may have improved - in which case someone can take credit for overall improvement. But there was no reduction of a relative racial performance gap, so anyone trying to take credit for that is a crook.
NC 1992-93
63.4% whites passed the treshhold. That means the whites' average score is 0.36 white Standard Deviation below the treshhold - using standard Z-table lookup.
30.1% blacks passed the treshhold. That means the blacks' average score is 0.5 black Standard Deviation above the treshhold.
AvgScore White = Treshhold Score + 0.36 wSD
AvgScore Black = Treshhold Score - 0.5 bSD
Delta Avg Score = AvgScore White - AvgScore Black =
Treshhold Score + 0.36 wSD - (Treshhold Score - 0.5 bSD) =
0.36 wSD + 0.5 bSD.
If the black and white SD were the same, adding them up would yield 0.86 SD difference between average scores.
Assuming black SD is 0.9 of white SD, 0.36 * 0.9 + 0.5 = 0.90 black Standard Deviations difference between the average scores of white and black samples.
Why the treshholds shift from case to case does not matter. Do the average scores shift? They would if the test was the same and the abilities of the students really improved. We have no idea and neither do we care since we calculate them relative to the pass treshold score - which we do not know and which gets eliminated from the calculation. All we know is that on each of the four independent tests I've analysed a score that is bested by 50% of whites is only bested by 16-18% of blacks.
miko
