I agree with you, but you are using the wrong numbers.

Sure, only 23% of households have 4+ members, but they contain 44% of all Americans. Likewise, the 1.3% of households with 7+ members contain 6.2% of Americans.

And there are more people in households of 6+ members than in single-member households.

  #people_in_household  %_of_households  %_of_people
  1                     28%              11%
  2                     34%              27%
  3                     15%              18%
  4                     13%              20%
  5                      6%              12%
  6                      2.2%             5.3%
  7+                     1.3%             6.2%

Collapsed to your categories:

  #people_in_household  %_of_households  %_of_people
  1-3                   77%              56%
  4-6                   21%              38%
  7+                     1.3%             6.2%

(source: https://www.census.gov/data/tables/2016/demo/families/cps-20... where tables C1 and A1 give a total 2016 population of 318579k. Table H1 gives numbers of households with 1-6 members, which, by multiplication, hold 298913k people. The remaining 19666k people must be in 7+ member households. This matches table AVG1's 2.53 people/household and surprisingly implies that the 1.6M households with 7+ people contain 12 people on average.)

And how about the purchasing power of a family of 1?

Is the purchasing power of a family of 5 (2 adults, 3 children) any higher than someone living alone, specifically when talking about purchasing home appliances?

I mean occasionally my wife and I will both purchase bread on the way home resulting in us having a lot of bread, but I can't think of a time we both accidentally bought a refrigerator or oven on the same weekend.

Well, a decent chunk of people living alone are probably renting, so that changes the whole appliance-purchasing dynamics already, but to directly answer your question - yes? The increasingly common dual income family has more income than a single person, and furthermore benefits from the ability to share certain costs efficiently - how would that not result in higher purchasing power?

Statistics for disposable income by family size is hard to get (I don't think the US census measures that). My gut feeling is that dual income, no kids families are going to be at the very top because not only are kids incredibly expensive, but not having kids means that you can concentrate on work. Having said that, family size will also skew around age so it's really hard to say. I'll just say that I've never met a family with kids that isn't concerned about money.

But even having more money, will people buy robot appliances? Again, my gut feeling is that singles are actually more likely to buy things like that because the downside of making an ineffective purchase is dramatically lower.