We live in a world in which, relatively speaking, we take rulers and tape measures for granted. They are useful things to have when you want to know how far one thing is from another. Rulers are made as a rule [sic] from wood or plastic, have engraved markings, and are rigid. Tape measures may be made from very flexible plastic or plasticised cloth, which can be rolled up tightly – these are often associated with sewing or knitting, and tend to be a bit stretchy – or a thin steel strip which winds up and retracts when not in use, usually into a small container under spring tension. Tape measures of both sorts usually have their division markings printed on them, and are almost invariably longer than rulers. The steel strip variety has an annoying habit of winding up unexpectedly, cutting your fingers in the process.

The Scales

The unit markings on the ruler or measure will show either or both:

• Measurements in feet, with sub-divisions for inches, half inches, quarter inches and perhaps eighths and sixteenths of an inch as well; precision¹ rulers for engineer types might also have 1/32 and 1/64 divisions.
• Measurements in metres², with sub-divisions for centimetres and millimetres.

If you want to measure, for example, the width of the path leading to the front door of your house, you get a tape measure (a ruler is probably too short), place one end of it at one side of the path, then extend the measure across the path, trying to keep it as close as possible to a right³ angle. Once the measure is in place, you read off the distance on the measure where it intersects the other side of the path. This may require two people if the distance being measured is more than the distance between the fingers of your outstretched arms. However, various cunning ways of doing this unaided have been developed by males when the distaff side has gone shopping just when you need her.

Wot, No Markings?⁴

So all well and good, but supposing the planet you are currently living on has never got round to small divisions, and only has measuring sticks calibrated to the nearest local equivalent of the inch or the centimetre? You need to be more precise. What do you do then?

Well, you could guess. Is it less than halfway between two adjacent marks? Greater than halfway? Or somewhere about in the middle? This roughly doubles your measurement precision⁵, but you need to do better, say 10 times better. Wow! Using the 'guess' method, you know that the width of the path in question is between the 134 and 135 marks, and appears to be a bit less than halfway between the two.

Fear not, there's a handy branch of mathematics ready and willing to come to your aid – statistics. Now, statistics is often recognised as having twin siblings – Lies and Damned Lies⁶, but it also has a magical cousin whose parents are Science and Reason.

And What Do We Do Next Jim?⁷

As always, design of the experiment is key. For this exercise, you will need something stiff, straight, and at least 50 percent longer than the width of the path you want to measure. The extra length is necessary to give a reasonable overhang on both sides of the path – be patient, you'll see why soon enough. If the local DIY⁸ store doesn't have a measuring stick long enough, you can always make one by cutting a piece of square-section timber and calibrating it yourself; it takes longer but works just as well.

The next stage is critical to the successful outcome of the exercise. You need to stand on the path, hold the stick horizontally in your hands – and casually drop it, perhaps whistling nonchalantly as you do so. This step (the casual dropping, not the whistling) is so critical that you need to repeat it several times, the more times the better really. Each time you drop the stick, check that it overhangs both sides of the path, and that it is reasonably close to a right angle to the side – if it isn't, you can nudge the stick with your foot, so long as you don't inadvertently introduce a bias by, say, consciously making one of the marks line up with the edge of the path. The other thing you must do each time is to note down carefully two numbers: the number on the stick nearest to the left side of the path and the number on the stick nearest to the right side of the path. When you have done this several times and noted down the relevant numbers, you can put them down in the form of a table (see below). This will enable you to both measure the path and to produce evidence to persuade the growing crowd of neighbours that you are not, as they are now convinced, barking mad.

The Table

The table should have three columns: one for the values for the left side of the path (L), one for the values for the right side (R) and one for the... well, we'll come to that in a bit. Let's assume you managed to drop the measuring stick 10 times⁹ before the crowd starts to look menacing. Your table should resemble this one:

To be really impressive, you should now put on your pointy wizard's hat before proceeding to the next stage, perhaps whispering magic incantations like Normal Distributions and Gaussian Curves. For each row in the table, subtract the value in the L column from the value in the R column, and put the difference in the third column that we left empty, like this:

Now we are ready for the simple bit of arithmetic that the magical statistics tells us we should apply. Add up the numbers in the R-L column – in our example table this comes to 1,344 – and divide by the number of rows - 10 in our case¹⁰. The result is 134.4. Therefore, the width of our path is 134.4 of whatever units our measuring stick was calibrated in: inches, centimetres, microfurlongs and so on. Although the stick is only calibrated to the nearest whole number of units, by using it in combination with statistics we are able to improve its precision¹¹ by a factor of 10.

Try it for yourself; it works. The secret is in the dropping of the stick. It introduces the random element that enables the statistics to do its stuff. Magic!