Question

When might one be interested in the mode or the median rather that the average? When might one be interested in the range rather than the standard deviation?

Answer 1

For example consider the case where different parts of rod are exposed to different temperatures(say 0 degree celsius and 100 degree celsius), it shall not be wise to consider mean as it will give 50, while median will make more sense. Also when the values taken by the random variiable are categorical good, okay and bad (denoted by 1,2 and 3 respectively) taking mean does not make sense because that will be in some way adding good and bad which are categories, in this case median or mode will make more sense. Also median and mode are more robust to outliers compared to mean( which gives equal weightage to each data point).

In some case one might be interested in knowing difference between highest and lowest value taken by the random variable rather than dispersion. For example the range of temperature witnessed by a particular city or the range of the marks scored by students in a particular exam. Also if X follows Unif(a,b) the random of random variable is more relevant is estimating a and b rather than standard deviation.