Question

When summarizing scale variable data using descriptive statistics, what do we lose in our understanding of a sample if we just report the mean and not any measures of dispersion like range or standard deviation? How does this question apply to a variable like age or test scores?

Answer 1

When summarizing scale variable data using descriptive statistics, if we just report the mean and not any measures of dispersion like range or standard deviation, we lose the extent of variation of the data points around the mean value in our understanding of a sample. The mean gives just the expected value of the given data set. But, our understanding of a sample requires the study of the extent to which the data points are scattered around the mean value also. This question applies to a variable like age or test scores as follows:
Suppose we are given a raw data of ages of all students in a class. The mean gives just the expected age of the students of the class. But our understanding of a sample requires the spread of the values. If the measures of dispersion like range or standard deviation is very small, then we can conclude that the ages of the students are almost consistent without much variation from student to student. If the measures of dispersion like range or standard deviation is very large, then we can conclude that the ages of the students are almost inconsistent with much variation from student to student.

Suppose we are given a raw data of scores of all students in a class. The mean gives just the expected score of the students of the class. But our understanding of a sample requires the spread of the values. If the measures of dispersion like range or standard deviation is very small, then we can conclude that the scores of the students are almost consistent without much variation from student to student. If the measures of dispersion like range or standard deviation is very large, then we can conclude that the scores of the students are almost inconsistent with much variation from student to student.