Unequal battle

The GINI equality index. Green good, Red bad...

When I first went to China in 1976 it was a very egalitarian society.  Poor, but egalitarian.  An average worker made about $US 10 a month (!), while a boss might make $15 or $20.  Blue and white collar workers made about the same.
Nearly 40 years later China is a much more unequal society.  It's a lot richer too.  There are bagloads of billionaires, millions of millionaires and a rising middle class.
So, is it "bad" that China's become more unequal?
Ask any Chinese, especially those that can remember the old times and it's a resounding "no", it's not a bad thing, it's a good thing; even if we might wish for a bit more equality, they may say, if that's at the expense of development and getting richer, then more inequality is fine. I know they have this attitude because when I visit, I always ask.  (scientific, that..).
Now, there's heated debate on this issue of inequality vs more equality.  To the Left, inequality is innately a Bad Thing, and more equality is a manifestly a Good Thing. (e.g.: Crooked Timber and Paul Krugman)
I wondered, if a more equal society is a Good Thing, there ought to be a correlation between equality and a measure of that good; let's make it "happiness".  More equality ought to = more happiness.
The best measure of equality is the GINI coefficient, using the most commonly accepted World Factbook figures.  Scores are between 0 and 1, with zero being the most equal (everyone gets exactly the same income), and 1 means that one person gets all the income and everyone else gets nothing.
The measure of "Happiness" I used was this index.  Sure it's called a "Prosperity Index", but then it does include 8 sub indexes: Economy, Entrepreneurship & Opportunity, Governance, Education, Health, Safety & Security, Personal Freedom, and Social Capital.  So, comprehensive, and not a bad proxy for "Happiness", I reckon.
And what are the results?
No significant correlation.

Spreadsheet here.

In short, only 17.5% percent of the variation in "happiness" can be explained by variation in income equality.
Another thing to note: for the top 20 "happiest" countries in the world, which happen also to include most European countries, Australia, New Zealand and the US, equality has actually improved (that is, the average GINI index has gone down, by 2.2%).  That's in contrast to what one commonly reads that there are widening -- scary, unacceptable, etc -- gaps in income inequality.  (The US is admittedly the one exception, having increased its GINI index by 10%).
I don't think I'd argue from this that income inequality doesn't matter. But maybe it only matters at the extremes: you can be "too equal" (China of the 70s, today's North Korea), and you can be "too unequal" (Marie Antoinette's France?).
Then again, Deirdre McClosky does argue that it doesn't matter.  Her rebuttal of the radical egalitarian du jour, Thomas Piketty, he of the "worlwide tax on the rich" fame.