Question

a) Explain intuitively why denseness is a desirable property of a sample set in sampling-based planning.

b) Explain intuitively why we want a low dispersion sample set in sampling-based planning.

Answer 1

Both part a and b is the same question as more denseness means low dispersion. So let me answer the 1st question which also works for part b.

To clear the concept intuitively let's consider a scenario.

Suppose you want to cook food for students of your class. You have budget constraints and so you have to ascertain the amount of food to make so that very less or nothing get spoilt. Hence you are interested in knowing the mean eating capacity of all your classmates and so take a random sample of students from the population of all the students of that particular class.

We already know that random sample is a good representative of the population units. So if you find that eating capacity of all students in your random sample is equal to say "k" bowls, you can infer that in total, all of the students may eat close to k*N bowls and thus you cook something more than k*N bowls where N is the total number of students in your class. This is the case with absolutely dense sample set or ideally zero variance sample set.

But suppose if eating capacity of students in your sample range from 2 bowls to 20 bowls (say), can you ascertain what amount of food shall be optimum for the whole class? This is a case of very large variability in sample set.

Thus intuitively it's clear that low variability/denseness is desirable in sample set.

[NOTE: Mathematically speaking, efficiency of any estimator based on random sample is inversely proportional to the sampling variance of the estimator. So if variance is low in sample then the variance of the estimator will be less, thus increasing the efficiency of the estimator.]

Hope the answer helps. Thank you.