Shown are the exam scores of 10 students. Calculate the mean score. Calculate the median of the scores. Then explain which of these measures of central tendency better represents a typical score in the exam.
Let’s start with the mean score. We find the mean score by taking the sum of all the scores and dividing that by the total number of scores. We know the exam scores of 10 students. The total number of scores then is 10. Our numerator should then be 76 plus 63 plus 77 plus 71 plus 81 plus zero plus 54 plus 93 plus 47 plus 82. When we add this together, we get 644. And that value is still over 10. 644 divided by 10 is 64.4. And so, we can say that the mean score of this exam is 64.4.
We move on to finding the median. To find the median, we need an ordered list. We either need to order our list from least to greatest or from greatest to least. Moving from least to greatest, we notice that there was a score of zero. And the next highest score was 47 followed by 54, 63, 71, 76, 77, 81, and 82, and finally 93. Since we have an even number of scores, the median is going to fall halfway between 71 and 76. There are five scores to the left of this value and five scores to the right of this value. And so, we take 71 and 76, and we find their average.
The median will be equal to 71 plus 76 divided by two, which equals 73.5. We could put that value here, 73.5. Because 10 is a small number, it’s easy to figure out where the middle value would be. We also have a formula to help us locate the median. And that is 𝑛 plus one divided by two, where 𝑛 is the number of terms you have. We had 10 terms. We would say 10 plus one equals 11, and 11 divided by two equals 5.5. This tells us that the median will be located halfway between the fifth and sixth term.
Now we’re ready to answer the third part of our question. Explain which of these measures of central tendency better represents a typical score in the exam. To answer this, we need to consider how the median and mean score compare. The median score is nearly 10 points higher than the mean score. This is because there was a score of zero. This score of zero has heavily affected the average. That is, the mean. In this case, we would say that the median is actually a better measure of central tendency because the mean is skewed too low due to the score of zero.