Yes, and in our experience, though often complicated for many viewers, Google Analytics provides some of the best analysis of website rank and SEO results. We have used many over the years, and once Google came along, it was clear the attempt was to give more relevant details. That being said, the most important thing to understand is, there is more to understand than just traffic increases.
If a business has a website that wasn't optimized, it's likely that when a little minor optimization takes place, there will be some increases. That doesn't mean that the SEO company is doing great (they could be), it simply means that things were so bad before, you are now starting to get traffic because you are actually doing some search engine optimization. This is why, like it or not, SEO should be a regular part of any company marketing budget. The main differences between simply optimizing a website, and a good SEO agency has more to do with consistent results, and long term growth.