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A Combination of Google APIs and scraped data.
We save all available data to our DB for it to be available after Google shuts down dislike counts in their API.
Right now video dislikes are cached, and aren't updated very frequenly. Once every 2-3 days, not more often.
Yeah, it's not ideal, but it is what it is. Working on improving how often we can update them.
The extension collects the video id of the video you are watching, fetches the dislike (and other fields like views, likes etc) using our API, if this is the first time the video was fetched by our API, it will use the YouTube API to get the data, then stores the data in a database for caching (cached for around 2-3 days) and archiving purposes and returns it to you. The extension then displays the dislikes to you.
The backend will switch to using a combination of archived dislike stats, estimates extrapolated from extension user data and estimates based on view/like ratios for videos whose dislikes weren't archived and for outdated dislike archives.
RYD uses the votes from its users to extrapolate the dislike count.
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If the video was uploaded after the API was shut down:
$$ \textup{RYD Dislike Count} = \left( \frac{\textup{RYD Users Dislike Count}}{\textup{RYD Users Like Count}} \right) \times \textup{Public Like Count} $$ -
If the RYD database somehow had the actual like and dislike count (provided by the uploader or from the archive), the dislike count will be calculated based on both - the users' votes and the archived value. The archived value will have less influence on the final count as it ages.
This in video form
See this page for more info.