Why is statistical dispersion important?

Answer 1

If we will talk about SCIENCE, then “an idea” is quite a scanty gift.
The statistical dispersion is important because it is a measurable quantity.

In Descriptive Statistics (presentation, quantifying or summarizing data), the statistical dispersion is a measure of the statistical variability. The example shown by Allen is correct, although somewhat qualitative.
It should be added that dispersion is a measure of the ERROR in series of measurements.

However, it’s not the only goal of STATISTICS as a science.

The most important designation of Statistics is to make:
* PREDICTIONS of results
** REJECTIONS of HYPOTHESES.
These kinds of methods are presented by so-called Inferential Statistics (or Statistical Induction).
Without knowing the dispersion value, we could not be able to predict the probability of an event or to choose the more probable hypothesis.

Answer 2

It gives you an idea of the variability of your data. If you do an experiment in science lab, for example, and you do the experiment 5 times, and every time, you get the measurement "5", the average is 5, and there is no dispersion. Now imagine you do the experiment, but get the answers 1, 3, 5, 7, 9. The average is still 5, but look at that spread. How confident are you NOW about the measurement?