Which Measure of Average is the Best to Use?
Date: 08/17/2006 at 04:54:22 From: brian Subject: Finding AVERAGE of internet bandwidth speed I have collected over 120 observations of broadband internet connection speeds achieved through my service provider. The values range from 5 kilobits per sec. to 256 kbps. I need to determine the average speed. I fed these numbers on the www.wessa.net site and got the following results: Arith. mean 75 (S.E. 69) Geometric mean 44 Harmonic mean 22 Quadratic mean 102 Winzorized mean 0/123 75 40/123 56 Median 50 Midrange 129 Midmean 54 Earlier, we used to make use of harmonic mean to determine the average speed of a vehicle, for example. I am confused as to which measure would be nearer the reality, even if a single value may not capture the normal distribution. Thanks in advance for any thoughts you may have that will help me take up the issue of low speeds with my service provider, in a way that they will understand.
Date: 08/17/2006 at 14:24:33 From: Doctor Douglas Subject: Re: Finding AVERAGE of internet bandwidth speed Hi Brian. It all depends on exactly what you mean by "nearer the reality". In the case of a dispute with your provider, I would probably make your argument along the lines of: 75% of the time, the speed was less than X kbps, which is unacceptably low. The reasoning behind this is that presumably very high connect speeds are nice, but you may not really care about that--they don't really offset the low speeds (i.e. below some cutoff threshold that you choose) which prevent you from using your connection. It is the distribution of speeds (particularly low ones) that is the crux of the issue. The mean or median has only minimal information about this. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
Date: 08/19/2006 at 02:35:59 From: brian Subject: Finding AVERAGE of internet bandwidth speed Thanks a lot. Your comments are very helpful. The histogram of my data in 7 groups shows: min. speed 2.1 kbps 45.53% 38.4 17.89% 74.8 9.76% 111.1 7.32% 147.5 8.13% 183.9 6.50% 220.2 4.88% So, I suppose that the argument could be that they are able to give near to the assured speed of 256 kbps for less than 5% of the time (the last group). Just one small question though: Why is the harmonic mean NOT a good indicator? Thank you for your thoughts and guidance. Regards
Date: 08/19/2006 at 20:25:38 From: Doctor Douglas Subject: Re: Finding AVERAGE of internet bandwidth speed Hi Brian, The harmonic mean IS a good indicator for some purposes when you are trying to find an average rate. For example, suppose you are on a driving trip, and you travel 100 miles at 40 mph, 100 miles at 50 mph, and 100 miles at 60 mph. What is your average rate for the entire 300 mile trip? rate = total distance / total time = 300 mi / (100/40 + 100/50 + 100/60) = 300 mi / (6.167 hr) = 48.65 mph, which is the harmonic mean of 40, 50 and 60 mph. So you see that the harmonic mean is the appropriate average to use when you perform the same task ("100 miles" in the above example, or perhaps in your original context, "download a 20 MB file") at each of the individual rates. Now, what if the driving trip consisted of 2 hours at 40 mph, 2 hours at 50 mph, and then 2 hours at 60 mph? Then the average rate for the entire trip is rate = total distance / total time = (2*40 + 2*50 + 2*60) / 6 hr = (80 + 100 + 120) / 6 = 50 mph, which is the normal average of the three rates. So it's clear that the appropriate average depends on the situation. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
Date: 08/21/2006 at 00:27:06 From: brian Subject: Thank you (Finding AVERAGE of internet bandwidth speed) Thank you very much for your very helpful comments.
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