Wait, I did everything right and my page's average rank dropped?

Not only is it fairly common, it's a sign of growth.

As your improved page grows, it will start to rank for new keywords. Those new keywords will have lower ranks than what you're reaching on your more established keywords. That drops your average.

Here's a bike touring company that improved their page titled 3 to 4 Day Bike Tours. Originally, that page ranked for 4 keywords--three of them on the first page. Those 4 keywords ran an average rank of 7.25.
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After improvements, the page gained in all 4 of those keywords. It also ranked for new keywords that it hadn't before--coming in on the 2nd and 3rd pages for most of those.

In this example the page did not lose position on any keyword, but its average rank dropped from 7.25 to 16.

Had you measured only the original 4 keywords, your average rank would have gone from 7.25 to 3.5.

The page grew and improved. It jumped ranks on important keywords and gained more clicks and impressions. It's a success story.
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One way to gut check this is to spot the growth in your Impressions, Clicks, and Number of Keywords.

Ranks are only a step toward higher clicks (traffic) and impressions (brand exposure). Ranks don't earn you a dime, but traffic can convert into a paying customer. As you improve your content, be mindful of the value that you continue to build.

It's still important to track the individual ranks of your most important keywords. A commitment to ongoing content improvement will help you stay on top of significant rank changes, as they are an indicator of performance when they're singled out.