If you examine the USDA website, near the bottom of the page you'll find this:
The amount of benefits the household gets is called an allotment. The net monthly income of the household is multiplied by .3, and the result is subtracted from the maximum allotment for the household size to find the household's allotment. This is because food stamp households are expected to spend about 30 percent of their resources on food.Emphasis mine. Clearly, food stamps are not expected to fund the entire gocery bill. It might be a job, or welfare, providing most of the grocery money.
At the above link, you will also find a table showing the standard monthly rates for food stamps. The weekly amount of this supplemental assistance for a single person, before means testing, is $155 per month or $35.67 per week (divide by 4.345 - average weeks per month). $5 per day then, is about what food stamps are worth to a single person. However, $5 per day is definitely NOT what “the state’s average food stamp recipient spends on groceries.”
Food stamps are supplemental, and to pretend otherwise is disingenuous.
It is also worth looking at what the USDA publishes as a guide to the cost of a healthy diet for a frugal consumer. A single male, 20-50 years old, for example, can live on as little as $187.50 per month for food. In other words, food stamps might cover 83% of his basic need. Data here.
H/T Bizzyblog for the USDA links. Bizzyblog also reports on similar chicanery in other states.