h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

So why is it the mean phone call turns out to be the most generous?

Assuming that by generous you mean the longest. I have no figures to go on here but, the mean is skewed by the long length of voice calls. Although there are relatively few of them because they are so long compared to the other two call types mentioned they pull the mean value up.
There is another type of call, not mentioned in the article, that will also have this efefct. Internet calls are also usually quite long and will pull the mean higher. Voice and internet calls do not have much effect on the median because the number of failed calls and faxes is so great, the number of voice and internet calls is relativly small. They do not have an effect on the mode due to the large peak described in the article that is attributed to failed calls.
Wick

Exactly right.

A graph of call length against number of calls will have a high peak at the beginning (between 1s and 4s) and then fall away sharply.

There is then a peak as you get to around the time of a single page fax (rises again after about 30s and starts to fall off somewhere before a minute). Because many conversations are also around a minute this peak rises slower and falls off slower.

You will then see many calls that are over a minute. Half of the total number of calls will be more than around 45 seconds long meaning that they can't help but skew the mean upwards.

(If we were to take the following as 10 random calls you may see 2s, 2s, 10s, 35s, 45s, 50s, 55s, 1.30m, 1.45m, 4m. This will give a modal of 2s (2 x calls) and a median of 47.5s. The mean is 1.18m. It may well be that all of the calls between 35s and 1m (4 in total) were faxes, but because they are more spread than the two calls where somebody probably just heard the "Hello we're not in" bit of the anser message they won't make the modal. In a larger sample you will inevitably find many more longer conversations which will bring the mean up more.)

All of this assumes that you measure the call length to the nearest second of course. If you were to use the nearest minute then over half the calls would be in the first minute. This causes the modal and the median to both be in the <1 minute block, but the mean is still likely to be in the 2-3 minutes block.

Different carriers (telephone companies) will get different answers as some specialise in particular types of traffic, for example fax or long distance voice so their results could be quite different. The example is really for a carrier that does voice, but with a high percentage of business customers (home customers are far less likely to cause the median to fall nicely in the 35s-50s bracket as they don't normally send faxes).


K