March, 2002
Homework Assignment Number 4 had you do a "Fermi Number" problem. We have compiled all the answers, which I summarise here.
This is the only one of the three problems for which an answer is known. For non-balding people, the number of hairs on their head is between 100,000 and 150,000, depending on the individual, their coloring, etc.
| Number of Answers | 19 |
| Minimum | 2600 |
| Maximum | 2,000,000 |
| Mean | 191,990 |
| Median | 59,000 |
You may wish to know that the mean is another word for the average: it is the sum of all the answers divided by the number of answers. The median is the number for which one-half of the answers is less than the value and one-half of the answers is greater than the value. In situations like this, the median is usually a better measure of the "typical" answer. For this problem, the differences between the two values, 191,990 and 59,000, are not very great; below we will see an example where this is not true.
| < 10,000 | 3 |
| >= 10,000 and < 100,000 | 8 |
| >= 100,000 and <= 200,000 | 5 |
| > 200,000 and <= 1,000,000 | 2 |
| > 1,000,000 | 1 |
| Number of Answers | 18 |
| Minimum | 2400 hours |
| Maximum | 269,166 hours |
| Mean | 103,436 hours |
| Median | 93,280 hours |
| < 10,000 | 4 |
| >= 10,000 and < 50,000 | 3 |
| >= 50,000 and < 100,000 | 3 |
| >= 100,000 and < 200,000 | 4 |
| >= 200,000 | 4 |
My partner, son and I together got 100,000 in a discussion over dinner one night. This is not necessarily a "better" answer than yours! However, I think the minimum and maximum values reported below are clearly way off.
| Number of Answers | 23 |
| Minimum | 0.0162 m2 |
| Maximum | 3,700,000,000 m2 |
| Mean | 161,074,000 m2 |
| Median | 47,677 m2 |
Note the large difference between the value of the mean and the value of the median. Below I shall comment on this further.
| < 10,000 | 7 |
| >= 10,000 and < 50,000 | 5 |
| >= 50,000 and < 100,000 | 4 |
| >= 100,000 and < 200,000 | 2 |
| >= 200,000 | 5 |
The answers to this question provide an example of why I prefer the median to the mean: the one "wild" value of over 3 billion has seized control of the mean, making it unreasonably large. For example, imagine we have 10 measurements of some quantity:
| 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1,000,000 | 1.5 | 1.5 | 1.5 | 1.5 |
The mean of these 10 numbers is 100,001, while the median is 1.5. The "wild" value of a million has badly skewed the mean.
Here is a way to estimate the number of hairs on your head which I think is pretty nifty. Your mileage may vary.
Imagine dividing your head in two, the left side and the right side, and consider only, say, the left side. Now divide that side in 2 with approximately equal numbers of hairs in both halves. Divide that in 2, and so on until you end up with a small enough patch of scalp that you can actually count the number of hairs growing out of it. Say that in the process you have done N divisions. Then the number of hairs on your head is the number you counted times 2N.