If a data chart has greater variability, does that mean it has a greater range?

No! Just because it has a bigger variety doesn't mean that there is a greater range. There could just be a lot more numbers in the same category. For example, say that there was a category that was 0-5. There could be 352702375279305 in that category and it still wouldn't be a greater range, just a bigger variety. 

Hope I could help! 
-Marshy