In the first release, the index gave US President Barack Obama a score of 34 -- down four points from the previous day -- to 25 for Republican rival Mitt Romney, who was up two points.
But a historical look at the index since May 1 showed wide variation, with Obama's score ranging from 10 to 74 and Romney's from 14 to 64.
The index is calculated with a company called Topsy, which provides "real-time social analytics" by analyzing tweets to calculate a daily score measuring how Twitter users feel about the presidential candidates.
"Imagine getting a glimpse into election conversations happening all over the nation every day. Topsy analyzes massive amounts of tweets in real-time to measure those conversations, understand what people are talking about and make sense of it," said Duncan Greatwood, chief executive of Topsy.
"Through this partnership with Twitter, we're excited to make this the measurement to get the pulse of the nation leading up to the fall elections."
Topsy said it studies more than 400 million tweets a day in an effort to gauge how people feel about any key word or term related to the candidates.
"Thanks to this depth of data and scale of technology, Topsy's social sentiment algorithm has been tested to agree with human classification more than 90 percent of the time," a company statement said.
"Tweets are then given a score that ranges from zero to 100, with higher scores being more positive and lower scores more negative."
Adam Sharp, Twitter's head of government, news and social innovation, said the index "lends new insight into the natural, unprompted opinion of the electorate."
"Just as technologies like radar and satellite joined the thermometer and barometer to give forecasters a more complete picture of the weather, so too does the index we've partnered with Topsy on stand with traditional methods like surveys and focus groups to paint a more complete picture of the political forecast," Sharp added.
Topsy said that even though it uses different methods, its index is "highly correlated" to Gallup Poll's approval ratings for Obama for the past two years.