A normal distribution, or bell-shaped curve, is a theoretical model used to evaluate an empirical distribution of scores. Ideally, distributions should closely resemble the normal curve, with the median, mode and mean coinciding with each other at the exact middle of the distribution, and the scores should gradually decrease from the mean on each side of the middle, with approximately half of the scores falling below the mean and approximately half of the scores falling above the mean.

A test in which 75 percent of the scores fell below the mean, leads me to conclude that the scores are positively skewed and therefore not evenly distributed. In this case the mean is pulled in the direction of the high outliers above the mean, creating an inflated mean score.

The instructor should calculate the mean score and standard deviation, and adjust the distribution to fit the new normal curve so that 99% of the scores are within three standard deviations of the mean.

*Imagine that you recently took a statistics exam and your instructor just returned your graded exam. The instructor announces that 75 percent of students scored below the mean. How do you reconcile this with the fact that, in a normal distribution, half the scores should fall below the mean and half of the scores should fall above the mean? What descriptive statistics do you think instructors should use to evaluate whether or not to curve exam scores? Can you describe a method of curving that you think is statistically sound?*

That is a very elegant solution Sam. The results are definitely skewed, and quite heavily.

Sam,

I agree with you; for this example the data is not evenly distributed. That makes it difficult to have a bell shaped and symmetrical normal distribution. In my opinion, instructor can adjust scores or be selective in what particular scores to include in the distribution; which will make it easier for attaining normal curve distribution. I am not sure if this is a feasible or technically sound way to attain normal curve distribution but I do not see any other way.