Question regarding probability and normal distributions.

[cerealkiller]

Hello Mathematicians!

I'm doing some work on obtaining true measures of ability for students, and am trying to find a simple mathematical example that would show that a student's true ability is obtained by having a few equally weighted tests rather than one big test.

The example I'm thinking of is something along the lines of:

A student's "true" ability is 70/100, but if they sit an exam there will be a slight error in their ability measurement. Say that their mark will be normally distributed with a mean of 70 and a standard deviation of 5. So there is a 68% chance that their mark is between 65 and 75.

Now rather than sit a single exam, say that they sit 2 exams instead - both of which their mark will come from a normal distribution with mean 70 and standard deviation 5. The exams will be equally weighted as 50% of their total mark.

Now earlier, if they sat a single exam, there would be a 68% chance that their mark would be between 65 and 75. Now that they are sitting 2 exams each weighted at 50%, what would the probability be that their total mark is between 65 and 75?

Also, what about if there were 3 exams of equal weighting, etc.?

Thanks for the help guys.

 

[statdad]

So you have two scores T1 and T2, identically distributed, and want to calculate T = .5T1 + .5T2 and say something about its distribution?

The difficulty is that there is no way you can assume the two test scores are independent, since the same person takes the two tests. You'd need to make some assumption about the correlation between the two scores.

 

[FactChecker]

The difficulty is that there is no way you can assume the two test scores are independent, since the same person takes the two tests.

But it is exactly the dependence on the person taking it that the OP is trying to measure so that he can determine the student's ability. If the tests are spread out over several days, we can assume that the day-to-day variations even out. If the test results are low over several days (perhaps the student is sick for a few weeks), then we may have to conclude that the student is just less capable during that time.

I would say that dividing the test questions into several tests taken on separate days would definitely reduce the effect of day-to-day variations. I think that the central limit theorem could be used to show that the average error rate due to daily variations would approach zero as the testing is spread out over more days.

 

[statdad]

"I would say that dividing the test questions into several tests taken on separate days would definitely reduce the effect of day-to-day variations. I think that the central limit theorem could be used to show that the average error rate due to daily variations would approach zero as the testing is spread out over more days."

I doubt that splitting things into mini test would eliminate the dependence. The problem is that there is not enough information to allow us to model the form the dependence takes.

 

[blue_raver22]

Hello,

The concept of using multiple tests to obtain a more accurate measure of ability is known as the "law of large numbers" in statistics. Essentially, as the number of tests increases, the probability of obtaining a true measure of ability also increases.

In your example, if a student sits two exams with equal weighting, the probability of their total mark being between 65 and 75 would be higher than 68%. This is because the probability of obtaining a mark between 65 and 75 on each individual exam is 68%, and with two exams, the chances of obtaining a mark within that range on at least one of the exams is higher.

To calculate the exact probability, we would need to use the formula for the sum of two normally distributed variables. However, as the number of exams increases, the probability of obtaining a true measure of ability also increases.

I hope this helps clarify the concept of using multiple tests to obtain a more accurate measure of ability. Let me know if you have any other questions.