Cumulative Distribution of Income-to-Poverty Ratios in the U.S.

In August of this year, the U.S. Census Bureau published a report titled "Income, Poverty, and Health Insurance Coverage in the United States: 2006." Sadly, the report revealed that the number of Americans living below the poverty line and living without health insurance increased from 2005. According to the report, 12% of Americans (36.5 million people) live in poverty. Shockingly, 17.4% of children under 18 (that's 12.8 million children) under 18 currently live in poverty.

Of course, reporting the number of people below the poverty line doesn't tell the whole , tragic story. It's perhaps more revealing to look at the depth of poverty throughout the population. The depth of poverty can be measured by the "income-to-poverty ratio." If someone has an income equal to the poverty threshold, their income-to-poverty ratio is 1. If they make twice what the poverty threshold is, then they have an income-to-poverty ratio of 2. The text of the Census Bureau report mentions the number and percentage of people within three ranges of income-to-poverty ratios (those below 50% of poverty, those below 100% and those below 125%) but I was curious about what percentage of people are below other levels, like 150% or 200%. There is a figure that shows the approximate distribution of income-to-poverty ratios:



but it's hard to tell how many people might fall below a particular ratio. For that, it would be much better to have the cumulative distribution. So, I decided to generate the cumulative distribution myself and put it up here in case anyone else would like to see it:



Now we can look at this new figure and determine what percentage of people are below any arbitrary income-to-poverty ratio. Ordinarily, cumulative distributions go from 0 to 1, but in this case it doesn't start at 0 since there are some people with no income and it doesn't go all the way up to 1 because about 15% of Americans have an income-to-poverty ratio greater than 6. Interestingly, the curve passes through the 50% point at an income-to-poverty ratio of about 3. This means that half of Americans make less than 3 times the poverty threshold. Unfortunately, the Census Bureau has defined 48 different poverty thresholds, depending on how many total people and how many children live in the household, so it's hard to put a dollar value to this statistic. To get a ballpark number, let's use the weighted average poverty threshold for one person living by himself (~$10,500) to estimate income per person in the household. This will actually be an overestimate, since the more people who live in the household, the higher the poverty threshold gets, but the income per person that constitutes the poverty threshold actually decreases.

Anyway, using this estimate, we can say that half of Americans make less than about $31,500 per person in their household. If you live in a city like LA, San Francisco, Boston or New York City, then this is still barely a living wage.

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If you're curious, here's how I made the cumulative distribution plot:

Mathematically, it's simple to convert between a probability distribution function and a cumulative probability distribution function -- just integrate. However, here we don't have a function describing the distribution. Instead, I had to extract the coordinates of the pixels on the curve from the figure (using NIH's ImageJ). I also extracted the pixels of the extreme values on the axes, which I imported into Excel so that I could do a quick linear fit to get equations for converting the pixel coordinates of the curve to values of "income-to-poverty ratio" and "density". Finally, I imported the converted values into MATLAB and used a function called "cumsum" to get a numerical approximation to the integral of the density values.

You can download the data for the distribution and cumulative distribution in a MATLAB MAT-file here.