The White House claims that:
On average, full-time working women earn just 77 cents for every dollar a man earns.
Is it the case?
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It is not as accurate to claim that on average full-time working women earn just 77 cents for every dollar a man earns. A more accurate claim using the proper mathmatical nomenclature would be ...
In 2011, the median full-time year-round* working female age 15+ earned 77 percent of what the median full-time year-round working male age 15+ earned.
* A full-time, year-round worker (page 20 pdf, labeled page 12) is a person who worked 35 or more hours per week (full time) and 50 or more weeks during the previous calendar year (year round). For school personnel, summer vacation is counted as weeks worked if they are scheduled to return to their job in the fall.
The White House website is publishing that statistic that also shows up in their Fifty Years After the Equal Pay Act (FYAtEPA) report.
This progress notwithstanding, in 2011, the average woman still earned only 77 cents for every dollar earned by men.
The source of the claim, according to the FYAtEPA, comes from the Census Bureau
 U.S. Census Bureau. “Women's Earnings as a Percentage of Men's Earnings by Race and Hispanic Origin.” Historical Income Tables, Table P-40. (2011). http://www.census.gov/hhes/www/income/data/historical/people/.
Table P-40 notes that the 77.0 percent ratio is a comparison of the median full-time female to median full-time male.
(Based on median earnings of full-time, year-round workers 15 years old and over as of March of the following year. [...])
The median and average(mean) are not the same thing, but sometimes the word average is used when a median value is reported for a statistic, because median values have a central tendency and are less skewed by exceptionally high/low values. The more commonly used "average" (mean) full-time working woman's earnings from the Census Bureau, Table P-37. Full-Time, Year-Round Workers by Mean Income and Sex: 1955 to 2012. Using those figures from 2011, you would determine that a full-time year-round working woman earns on average 72.2 cents (49040 / 67913) for every dollar a full-time year-round working man earns.
There is a wide range of criticism over the 77 cent figure. Some reporters even noted the White House's own pay gap, where the average women earns only 91 cents of what the average man working at the White House earns.
This comparison of median/mean earnings is on a broad level and does not control for many factors that can be significant in explaining earnings differences. One factor is that women on average work fewer hours then men do.
Among full-time workers (that is, those working at a job 35 hours or more per week), men are more likely than women to have a longer workweek. Twenty-six percent of men worked 41 or more hours per week in 2012, compared with 14 percent of women who did so. Women were more likely than men to work 35 to 39 hours per week: 12 percent of women worked those hours in 2012, while 5 percent of men did.
There are more women in occupations with lower-pay, and less women in occupations with higher pay.
While women are more likely than men to work in professional and related occupations, they are more highly represented in the lower-paying jobs within this category. For example, in 2009, professional women were more likely (nearly 70 percent) to work in the relatively low-paying education (with $887 median weekly earnings) and health care ($970 median weekly earnings) occupations, compared to 32 percent of male professionals.
In 2009, only 7 percent of female professionals were employed in the relatively high paying computer ($1,253 median weekly earnings) and engineering fields ($1,266 median weekly earnings), compared to 38 percent of male professionals.
Women on average have less work experience than men (page 37).
on average, women at every educational level and at every age spend fewer weeks in the labor force than do men. The differences between men and women in labor force attachment are much smaller among those with a college degree or more education.2
If the wording were "paid" rather than "earned" then the answer is clearly "yes", and while both the left and right agree that there is a gap, there is disagreement on the magnitude, as evidenced by user1873's answer.
However, if the word "earned" remains, statistically, this is unlikely the majority cause. Thomas Sowell outlines the general case here: occupational choice, etc.
The vast plurality of the cause of the gap is experience. This has been detected by many studies over the decades.
There is some residual gap that cannot be explained by conventional causal statistics.
Some are trying to attribute the remaining gap to women's risk aversion relative to men's.