So if I want to generate values for lightness in a table the way Photoshop calculates its greys, I should use the .907 primaries but with a Gamma of 2.2?

Y = 0.2126*R^2.2 + 0.7152*G^2.2 + 0.0724*B^2.2

I'm still missing something, because I can't make sense of my resulting values.

I think I've cracked it now.  Most of the formulae you see online are missing the gamma.  The formula should be:

Y = (0.2126*R^2.2 + 0.7152*G^2.2 + 0.0722*B^2.2) / 772.4106

Professor Hoffmann,

Thanks again for your taking the time.

Changing the gammas absolutely changes the order of the values.  The distortion I was originally experiencing is due to the fact that the linear coefficients give an intense value (say 255,0,0) a lower Y than a moderate shade of the same hue (say 175,30,30).  I don't fully understand the science behind gamma correction, but mathematically, if we want to boost the values of intense hues relative to others, the exponent must be greater than one; using 1/2.2 makes the problem worse. 

For example, I have three values in my table: red (229, 0, 0); cardinal (178.6, 28.8, 52.8); and oxblood (122.1, 32.1, 41.5).  Red is clearly the lightest, oxblood is clearly the darkest, and desaturating programs across different software and monitors agree.

Ranked light to dark by the Average method, we get Cardinal>Red>Oxblood

With the linear .709 coefficients, we get Cardinal>Oxblood>Red

With a gamma of 1/2.2, we get Cardinal>Oxblood>Red

With a gamma of 2.2, we get Red>Cardinal>Oxblood.

The last formula works emperically for my data, but I suspect it should actually be a little smaller because my resulting data seems skewed to the extremes. 

My application is really pretty trivial--I'm just making a table of color names.  But now I'm caught up in the intellectual exercise.  You can take a look at my old draft (using linear .907 coefficients) here.

So my formula should be:

Y = 0.2126*R^2.2 + 0.7152*G^2.2 + 0.0724*B^2.2

and I am not supposed to use the inverse of the gamma (1/2.2) after all? 

This seems to put my names in the right order, but it skews my data very dark (about one third of my data points wind up with less than 20% lightness).

Is it safe to assume that my RGB values are gamma-encoded? My values are based on component averages of responses to digital color swatches, so they haven't been converted into RGB from somewhere else.

At this point I'm tempted to just desaturate the image in PhotoShop and manually eyedrop each color into my data.  None of the formulas we've discussed in theory match the function in practice-- I don't mean to be a pain about it, I'm just surprised that it's such a mystery.

Okay, that helps.  So then I can add all of the components and apply an exponent of 1/2.2 to get back to percentages (50%, in this case).

My final formula is then:

Y = (0.2126*R^2.2 + 0.7152*G^2.2 + 0.0722*B^2.2)^(1/2.2)

This will give me a scale to 255, obviously you can divide by 255 to get %.