## Calculating average power of a set of data

There are n people with masses mn . Each run up a given slope of height h in times tn. Gravitational accelartion, g, is constant.
Now there are 2 ways to calculate the average power of this group of people with 2 different results. The question is which way of calculating the average power (hence which result) is correct.
1st way: I calculate the power of each individual: Pn =mn g h/tn and take the average of these Pn. This is the average power of the group calculated using the first way: Pavg.1 = (P1 + P2 + P3 +…….+ Pn ) /n
2nd way: I take the average of n masses, mavg and the average of n times, tavg, then use the power formula and obtain the average power of the group using the second way: Pavg.2 =mavg g h/tavg .
When I do the two ways on the Excel I get slightly different results. Here is an example:
g (m/s2) h(m) m (kg) t (s) P=mgh/t (W)
9,81 2,24 51 7 7844,861
9,81 2,24 48 7 7383,398
9,81 2,24 46 4 4043,290
9,81 2,24 47 8 8262,374
9,81 2,24 51 5 5603,472
9,81 2,24 49 4 4306,982
9,81 2,24 50 6 6592,320
9,81 2,24 52 4 4570,675
9,81 2,24 54 7 8306,323
9,81 2,24 48 5 5273,856
9,81 2,24 47 8 8262,374
9,81 2,24 51 4 4482,778
6244,392 ←average of the individual powers
9,81 2,24 49,5 5,75 6254,464
average m average t power ↑ of averages

So which way (result) would be correct? By the way, plotting t against m and using the slope of the best fit line to calculate power is not appropriate here, as the slope comes out to be negative (-0.087) and the correlation is very weak (R2 = 0.017). I greatly appreciate your comments.
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 Recognitions: Gold Member Science Advisor Your second method is not really valid. Just take a simple example of two runners who do it once, each. (nonsense values to make it obvious) If one runner had mass 100kg and ran up 10m in 10s - power is mgh/t = 100*10*g/10 =100g The next runner weighed 11kg and ran up 10m in 1s - power = 11*10*g/1 = 110g Mean of those two values is 105g taking both together, the power would be (by your calculation) 111*10 *g/11 =101g The data is dominated by the larger value. What you are dealing with is Harmonic Means (you are dealing with 1/t) and you need to be careful when they come into it. There is a similar problem when working out average speeds around a circuit. If you move round a square course (side length d) with the speeds on each side being 1m/s, 2m/s, 3m/s and 4m/s and want to work out the average speed, the 'obvious' choice is to say (1+2+3+4)/4 = 2.5 but the real answer involves finding how long it takes actually to cover each side t=d+d/2+d/3+d/4 =2.25d then the speed is 4d/t = 1.78 In this case (and we know from experience) the distances travelled at slow speed will dominate the average speed. There is a limit to just how much you can 'make up', on the fast bits for the slow bits. That's the Harmonic Mean for you!!!! Watch out for it.

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