You can tell you've really become a part of the national discourse when the New York Post writes a sensationalist story about you.

That's what happened to Instagram Friday morning, when the Post seized on numbers from AppData that showed the photo-sharing service had lost a quarter of its daily active users in the week since its Terms of Service debacle. But the data was misleading for so many reasons, it would make a good case study for a statistics class.

First of all, the Post article was cherry-picking numbers. AppData's record of Instagram's Daily Active Users plummeted from 16.4 million on December 19 to 12.4 million today. But the number of people who used Instagram in the last month actually rose in the same time period, from 42.5 million to 44 million.

You could just as easily write a story about the 1.5 million people who opened the app in the last week for the first time in December. Nate Silver would not be impressed with either piece.

It hardly bears pointing out that the period in question covered the Christmas holidays, when traffic is generally light across the web (though this is not always the case for apps). It does bear pointing out that AppData is only able to look at numbers from users who've connected their Facebook accounts to Instagram, which is by no means a plurality of its users.

And while AppData told the Post it was "pretty sure the decline was due to the terms of service announcement," that whole kerfuffle quietened down when co-founder Kevin Systrom made his apology on December 20.

No wonder Instagram was able to so confidently refute the story. Said a spokesperson: "This data is inaccurate. We continue to see strong and steady growth in both registered and active users of Instagram."

A Big Side Order of 'Yes, But'

All of that said, the Daily Active User statistic should still give the company pause — just as an abnormally high blood pressure reading means you should probably check in with your doctor, even if it is an outlier.

Yes, it's an incomplete picture of Instagram users. But it does cover 44 million out of 100 million total users, which is a pretty massive sample size in any statistician's book, especially when the sample contains no inherent bias. If pollsters had canvassed a random 44% of likely voters prior to the Presidential election, not even Mitt Romney would have been able to argue with their numbers.

Yes, it was Christmas. But shouldn't that mean more people want to take more pictures of more shiny things — presents, trees, relatives, food — with their phones, especially the new phones they found in their stockings? Instagram's Daily User figure actually rose over Thanksgiving, which only offered the relatives and the food.

Flickr's photo-sharing app saw a 100% gain in Monthly Active Users over the same Christmas period, according to AppData. To be sure, Flickr is starting from a lower base, and has the novelty factor on its side for now. (It's also offering a gift to users: 3 months of Flickr Pro for free.)

But Instagram should be concerned that dissatisfied users appear to be clustering around one particular alternative. As most tech industry veterans will tell you, in this business, it pays to be paranoid.

SEE ALSO: 5 Lessons from the Instagram Debacle

Systrom should be watching like a hawk for signs that his apology didn't cut it, or got lost in the noise — after all, it did land late on a Thursday night. In any case, just because he apologized on December 20 doesn't mean users have heard about it by December 25. Most of his 100 million users aren't paying as much attention as you, dear tech news reader.

Many will have heard something second-hand or third-hand about a change in the privacy policy, something about selling photos to advertisers (which, for all Systrom's protests about user confusion, was precisely what the new terms of service would have allowed.) Some will have only heard about it the moment they gathered around the dinner table with relatives. Still others will have heard about the apology, but don't know whether to believe it.

Instagram's potential health problem, then, has nothing to do with statistics and everything to do with perception. The Terms of Service story went viral, but the apology didn't. The danger is that Instagram has been indelibly defined in the public mind by this story, and is on its way to becoming a butt of late-night jokes, a dinner-table meme — and yes, a whipping boy for the New York Post.

How can Instagram fight that perception and make up with disaffected users? Give us your take in the comments.

Image courtesy of iStockphoto, andrearoad