The range of the set of numbers $6, 9, 3, 12$ is $9$.

To calculate the range of a set of numbers, you subtract the smallest number from the largest number in that set. In this particular case, the numbers provided are $6, 9, 3,$ and $12$. First, we need to identify the smallest and largest numbers. Here, the smallest number in the set is $3$, and the largest number is $12$.

Next, we perform the subtraction of the smallest number from the largest number:

Thus, the range of this set of numbers is $9$. The range provides insight into how spread out the numbers are, serving as a measure of the difference between the highest and lowest values.

Understanding the range is beneficial in various real-life contexts, such as analyzing data sets to assess variability. For instance, when examining the range of temperatures over a week, a larger range would imply greater variability in the weather, while a smaller range would indicate more consistent temperature readings. In summary, the range is a straightforward yet effective method for quickly gauging the spread of a set of numbers.