# Is it correct to take a simple average of weighted averages?

Final edit: The answer is "yes, it's appropriate."

The specific problem I'm having seems to be a rounding error in Excel. See discussion on the accepted answer for the "proof" (but not a Proof, of course.)

Obviously, the unasked second question is "if yes, how do I do it correctly?"

Given my lack of luck in finding something similar online, I suspect I'm either making an error or using the wrong search terms. Or possibly both.

I have a number of hourly data points, in the form of (count, average time) which I've weighted into an hourly weighted average for four different sources.

These are then averaged into a daily average-per-source, and then those averages are averaged into a average-per-process (two different processes.)

I'm not worried about the validity of averaging out averages in this case - it gives me the sort of data I'm looking for - but I get different results depending on how I average them when it seems like they should be the same. This means to me that one of the methods is right, and the other is wrong.

Equations I'm currently using (note, they're being done in Excel, though I'm trying to make them at least readable):

Hourly weighted average:

• R3 = ( (C3*D3) + (G3*H3) + (K3*L3) + (O3*P3) ) / Q3)
• R4 = ( (C4*D4) + (G4*H4) + (K4*L4) + (O4*P4) ) / Q4)
• ....
• R26 = ( (C26*D26) + (G26*H26) + (K26*L26) + (O26*P26) ) / Q26)
• C, G, K, and O are the counts. D, H, L, and P are the times. Column Q is the sum of C, G, K, and O.

Daily average:

• S3 = (R3 + R4 + ... + R25 + R26) / 24
• Average of the hourly weighted averages.

Average-per-process:

• V7 = (S3 + S51) / 2
• Average of two daily averages (S51 is a cell calculated similarly to S3 above.)

What I'm not sure of is whether it's valid to simply average out the hourly weighted averages as I did (similar question for averaging the daily into the "per-process" one,) or if I need to redo the weighted average process for them (i.e. (( count for hour 1 * average for hour 1) + ... ) / count per day)