Question

Imagine that you recently took an exam for certification in your field. The certifying agency has published the results of the exam and 75% of the test takers scored below the average. In a normal distribution, half of the scores would fall above the mean and the other half below. How can what the certifying agency published be true?

Answer 1

Here, 75th percentile is equal to average (mean). This means, the median of the dataset is below the mean. We know that, for normal distribution, mean = median = mode. When mean > median, the distribution is right skewed. So, the distribution of scores in this exams does not follow normal distribution. It is right skewed distribution.

In a right skewed distribution (positive skewness), the values towards the right side of median are more spread out and has very extreme values. These extreme values are influencial while calculating the mean and have less impact on median or mode. In the above mentioned result, the 50% who scored below median are having scores less than median, but close to it. But the scores higher than median will more spread out and can have some scores that are much higher that the median. These scores are the ones that makes the mean larger than the median.