It's a good start, but it's based on total income. So, someone with two minimum-wage jobs adding up to (say) 60 hours per week will appear above someone with a job paying twice minimum wage who only works 20 hours a week. That's not what we're looking for if we want evidence that minimum wage is increasing the wages of many on the low end of the scale.

Unfortunately, this happens to be what the census bureau tabulates, which makes it hard to find the data we're actually looking for.