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### 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
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

- 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

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)

```
Associated Topics:
High School Statistics

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