Determining margin off error within experimental calculations

[Thundagere]

For a physics project, a few friends and I got ahold of a couple of balls that would record the time of an impact. Using this, we wanted to calculate g using kinematics to see how off we would be from the expected 9.81 m/s^2. Obviously, differences in altitude, air resistance, and experimental issues mean that we won't be exact, but I was surprised to find that the value of our data averaged 8.5 m/s^2, much lower. I thought that we would get at least 9.2. Two questions, firstly, what is the best way to present this data? A physics instructor suggested the following method for experimental calculations:
Choose 7 random values of g from your list w/replacement. Take the average of them. Do this 5 to 6 times, or as many as possible. Take the range of the samples. Your answer is then average of all samples±range
Can anyone think of a better method? I'm not sure which would be best, considering we're this off. I know people like to use actual value- experimental value ÷ actual value, but for this project specifically, I'd rather not use this. Thanks!

 

[mikeph]

I'd just calculate the arithmetic mean for the best estimate, and then use the standard formula to calculate the deviation... sqrt(sum(value-error)^2/N).

It doesn't seem right to base your method of analysis on the details of your findings. You can't say "we were slightly wrong, so let's take the median instead of the mean and present that". You're slightly wrong for a reason, and if you tinker with a result you're unjustifiably concealing whatever effect caused the deviation! If you're calculating g from the free-fall time then I'd be more surprised if you got 9.81, since there will always be an air resistance term slowing the descent.

 

[mfb]

To determine the statistic uncertainty, I would use the variation in the sample itself, as described by MikeyW (and divide by N-1 instead of N).

To determine the systematic uncertainty, it would be necessary to know more about your setup.

 

[Thundagere]

Could you elaborate on the formula for deviation? I've never used it before, so I'm not familiar with it.

 

[haruspex]

Clearly you should ask a stats instructor, not a physics instructor! You can see how flawed the suggestion by supposing you only had 7 measurements and see what the scheme tells you to do. The MikeyW/mfb suggestion is far more reasonable:
- take the average
- calculate the difference from the average for each of the N measurements
- square each of those differences and add them up
- divide by N-1
- take the square root
This gives you a fair estimate of the standard deviation of your experiment. The probability that the actual value is more than X away from the average you calculated can then be estimated by comparing X with the standard deviation (D). You can look the probability up in standard tables for normal distributions, e.g. http://www.math.unb.ca/~knight/utility/NormTble.htm . For that table, calculate Z = X/D. Read out the value, p, from the table for that Z. The probability that your error is more than X is then 2*(1-p). E.g., if Z = .31, you read out 0.6217, and the prob that the error is more than 0.31*D is about 0.76

 

[Thundagere]

This is pretty interesting! Definitely seems a lot more precise than what my physics instructor gave me.
Thanks a lot for all of your help!

It doesn't seem right to base your method of analysis on the details of your findings. You can't say "we were slightly wrong, so let's take the median instead of the mean and present that". You're slightly wrong for a reason, and if you tinker with a result you're unjustifiably concealing whatever effect caused the deviation! If you're calculating g from the free-fall time then I'd be more surprised if you got 9.81, since there will always be an air resistance term slowing the descent.

Mikey, could you explain in a bit more detail why this method doesn't work? I want to make sure I understand it fully.

 

[mikeph]

Are you referring to your experimental method or you analysis method? I don't know why the "split into 7 then take the average" idea doesn't work... to me, it just doesn't make sense and it's not something I'd tell anyone to do.