Superstorm Sandy damaged some 300,000 homes, shuttered countless businesses, and knocked out power across huge swaths of the state.

In addition to testing first responders and cleanup workers, the mess also posed a gargantuan math problem: Just how, exactly, should government calculate the cost of all the devastation and disruption?

For New York City, the accounting is critical, because Mayor Michael Bloomberg is now asking the Federal Emergency Management Agency to cover 100 percent of cleanup costs and calling on Congress to reimburse any additional spending that’s not covered by private insurance.

The initial tally was released by the Mayor’s Office on Monday: a whopping $19 billion. Of that, the city estimated it was on the hook for some $9.8 billion after reimbursements by FEMA and insurers.

The city arrived at its figures, according to the mayor’s office and economic analysis experts, through a mix of consultations with private-sector firms, reviews of industry data, and internal number-crunching, including some back-of-the-envelope–style multiplication.

On Wednesday Mayor Michael Bloomberg took his request for $15 billion in storm recovery aid to the U.S. Capitol, where he was flanked by New York Senators Chuck Schumer and Kirsten Gillbrand. AP Photo/Evan Vucci

The city’s data sources and assumptions appear to be solid, according to Adam Rose, an economics professor at the University of Southern California who has studied the costs of disasters. While relying on estimates rather than hard data isn’t ideal, Rose said, the city has few other options.

“People haven’t filed their insurance claims, and you don’t have information on what everybody’s got,” he said. “What can you do?”

But it’s also clear that New York’s cost estimates are far from precise. One analyst who advised the city noted that using different assumptions about the region’s lost economic output could have changed her numbers by some 25 percent, a difference that would have cut projected costs by as much as $1.5 billion.

“We took the most rigorous approach that we thought we could do with the information that we have,” said Marisa DiNatale, a director at Moody’s Analytics. “But certainly, this is not going to be exactly correct.”

Just how uncertain are the numbers? Bloomberg’s office Monday released its aid request for roughly $10 billion — some $5 billion less than the request for the city that was issued by Governor Andrew Cuomo just a few hours later. Bloomberg’s office later revised its numbers, and in a Washington press conference on Wednesday, the Mayor said the city would be seeking $15 billion, but didn’t give additional details.

The earlier breakdown released Monday by the mayor’s office is the only one so far to give a detailed accounting of local costs, including a chart showing insured losses and FEMA spending — money that the city is already guaranteed.

The additional $9.8 billion request by the city outlined in that document boils down to three distinct categories: lost output, direct uninsured losses by private citizens and businesses, and direct losses and spending by city agencies.

The lost output makes up the lion’s share of the cost, at an estimated $5.7 billion based on Moody’s calculations. The number is an attempt to capture indirect storm-related costs like lunches that weren’t sold, or products that weren’t manufactured, as opposed to direct damages like flooded basements or totaled cars.

Moody’s calculated its loss estimates in the days immediately following the storm, according to DiNatale, who worked on the project with three of her colleagues as an independent analysis. The Moody’s team was later approached by the Bloomberg administration as it readied its request to Congress.

The Moody’s model, said DiNatale, began with the assumption that Sandy essentially ground all industries to a halt for two full days, then worked backwards to see which sectors would be able to recoup their losses. Manufacturing would likely fare well, since factories could ramp up production in the days or weeks following the storm to make up for lost time, while businesses like restaurants would probably struggle more.

“It’s not as if people were going to say, ‘I didn’t go out to eat Monday, so I have to go out to eat twice this week,'” DiNatale said. “That’s lost forever. In certain industries, you’re going to be able to make up that output, and others, you’re not. We tried to go industry by industry, and tried to calculate the percentage that was gone for good.”

After weighting the industry figures based on each sector’s importance to the overall economy, Moody’s pegged lost output at $5.9 billion in the five boroughs, which was actually $200 million higher than the city’s estimate of $5.7 billion. (The Mayor’s office did not respond to multiple requests for comment.)

DiNatale said that had Moody’s used different assumptions, its estimate could have run as low as $4.5 billion. But, she said, her team wanted to show what would be the “possible upper range.”

The remainder of the city’s costs consists primarily of private uninsured losses, which amount to some $3 billion, according to Bloomberg’s figures.

To arrive at that figure, the city first surveyed loss estimates from eight different insurance and risk management companies.

Those companies typically use models to project losses after hurricanes, which factor in inputs like property values, storm strength, and structure vulnerability, according to Kevin Long, the media relations manager at AIR Worldwide, one of the firms that the city relied on.

While the models are sophisticated, Long said that his company’s estimates are still projections; they can’t be based on actual claims, since filing deadlines haven’t been reached yet.

“Our losses are estimates,” said Long. “We simulate the event.”

The city used the companies’ data to assess the total value of hurricane-related insured losses across the country, then used numbers from AIR Worldwide to determine the proportion of those losses that would fall in New York.

Then, using a formula derived by a Swiss reinsurer, the city multiplied its estimate of insured losses by 125 percent to arrive at the value of local uninsured losses, which it assumed were split half-and-half between residential and commercial properties.

The city’s formula, Rose said, put the share of losses in the “normal range” for a major catastrophe. The total was $4.8 billion, but because FEMA pays for 75 percent of any residential losses, the city estimated that the non-reimbursed share of the cost would be reduced to $3 billion.

Finally, Bloomberg’s release showed an additional $4.7 billion of losses and spending on response and recovery by city agencies and the Army Corps of Engineers, of which FEMA would pay some $3.6 billion — leaving the New York on the hook for $1.1 billion more.

The city’s Office of Management and Budget and state Division of Homeland and Security and Emergency Services were responsible for tallying those figures, according to the Mayor’s Office — but they didn’t show what went into their calculations.

A footnote explains circumstances made it impossible to deliver an accounting of agency-by-agency costs: “A more detailed breakdown was not available from individual agencies as of November 21, 2012.”