One example in which the mean, median, and mode can be used effectively to reduce groups of data and illustrate measures of central tendency and variability is in a distribution of test scores. Describe the different ways in which the mean, median, and mode contribute to the interpretation of data. Be specific. Give an example in which these numerical descriptors of data can be effective (e.g., test scores).
The mean of a set of test scores tells you what the average score on the test was for the class.
The median of a set of test scores tells you what the middle grade is when you order the grades from highest to lowest. The closer the median is to the mean, the more normally distributed the data is.
The mean is used to interpret data when the distribution is normal, but for skewed data (say for instance most people did well, but there were several students who scored very low) is a more accruate representaton of how the class did as a whole. If several people in the class received zeros, the mean will be skewed towards the lower end of the grades.
For example, say these were the test scores for 10 students out of 100:
10 10 12 71 75 78 81 82 85 89
The mean is: 59.3
The median is: 76.5
In this case, three students skewed the distribution by scoring poorly in comparison to the rest of the class; therefore, the median is a better representation of the average performance rather than the mean.
The mode represents the grade that was obtained the most by students. As with the median, the closer to the mean it is the more normally distributed the data is. The mode is the most accurate representation of group performance when the data is highly skewed.
For example, here are the test scores for a different set of 10 students:
10 12 62 64 69 70 91 91 91 91
Here the the mode is 91, the median is 69.5, and the mean is 58.2.
Almost half the class received 91 on the test, and so therefore the mode is the best representation of the average performance of the class compared to the median and mode.
Mean - normally distributed data
Median - moderately skewed data
Mode - highly skewed data