$$\require{cancel}$$

# 6.28 Counting Statistics

Every measurement is made up of a number, a unit of measurement, and an uncertainty or observational error. The uncertainty is an integral part of the measurement; there is no such thing as a perfect measurement in science. Actually, the word "error" is a bit misleading, since no mistake has been made. Uncertainty is a better term, but we will stick to standard terminology. The random error, or scatter, in measurements, goes down as the number of independent measurements goes up. In fact, the standard error is proportional to 1/ √ N, where N is the number of measurements that were made.

This mosaic covers part of the equatorial region of Jupiter's moon, Callisto. The mosaic combines six separate image frames obtained by the solid state imaging (CCD) system on NASA's Galileo spacecraft during its ninth orbit around Jupiter. Click here for original source URL.

What happens if we are just counting things? In this case, there are no units, because a count is a pure number. Simply:

Count = N

The image shows a 120 km wide crater on mars, with subsequent impacts at later epochs within it. Evidence of these subsequent impacts occurring over large timescales is shown by some of the craters being buried. Click here for original source URL.

If the items being counted are governed by a random process, the counting error is given by:

Random error = √ N

Cutaway of the interior of the Earth showing different layers. Click here for original source URL.

The fractional error is therefore:

Fractional error = Random error / Count = √ N / N = 1/√ N

Enchanced-color image of Deimos, a moon of Mars, captured by the HiRISE instrument on the Mars Reconnaissance Orbiter on 21 Feb 2009. Click here for original source URL.

The correct way to quote a counting measurement with the error attached is:

Color image of?Phobos, imaged by the?Mars Reconnaissance Orbiter?in 2008. Click here for original source URL.

Count = N ± √ N

Full Moon photograph. Click here for original source URL.

In other words, the true count could be anywhere between N+ √ N and N- √ N.

Full-color image of Mercury from the first MESSENGER flyby. Click here for original source URL.

Counting errors only apply to events governed by random processes. There are nine planets in the Solar System, but we wouldn’t quote this measurement as 9 ± 3, implying that sometimes we might count 11 planets, and sometimes we might count 6. We have located all the planets in the Solar System, and there is no random uncertainty as to how many are out there. Some numbers are simple counts with no error attached.

Global mosaic of 102 Viking 1 Orbiter images of Mars taken on orbit 1,334, 22 February 1980. The images are projected into point perspective, representing what a viewer would see from a spacecraft at an altitude of 2,500 km. At center is Valles Marineris, over 3000 km long and up to 8 km deep. Note the channels running up (north) from the central and eastern portions of Valles Marineris to the dark area, Acidalic Planitia, at upper right. At left are the three Tharsis volcanoes and to the south is ancient, heavily impacted terrain. Click here for original source URL.

Counting statistics can apply to microscopic phenomena. For example, if a sample of radioactive material was measured to have experienced 285 decays within a fixed interval of time, the error in this count would be √ 285 = 17. We quote this measurement as 285 ± 17. The fractional error is 1/17 = 0.059, or about 6%. If we waited ten times longer, the radioactive material might yield 2792 decays, with a statistical error of √ 2792 = 53. We quote this measurement as 2792 ± 53. The fractional error is now 1/53 = 0.019, or about 2%. Notice that as we accumulate ten times more events, the fractional error has reduced by a factor of about √ 10 = 3.2, from 6% to 2%. The standard error of multiple observations is reduced by a factor of √ 10 if we collect ten times more observations.

With 100 random events, the statistical uncertainty is √ 100 = 10, or a 10% precision. With 10,000 events, the statistical uncertainty is √ 10,000 = 100, or a 1% precision. With 1,000,000 events, we have a 0.1% precise measurement, and so on. For a very small number of events, the uncertainty becomes large. For a single event, √ 1 = 1, and the math confirms what we already know: we can learn very little from a single random event!

Science often operates at the limit of counting errors. Astronomers collect individual photons of light with telescopes and try to accumulate enough counts to drive down the errors. We get plenty of light from the planets, so the limitations of light detection usually only apply to objects in the outer reaches of the solar system and beyond.

Picture of the lunar surface facing the Earth. Click here for original source URL.

Within the solar system, impact cratering is an important random process. Try counting craters on an image of the heavily cratered lunar highlands to see if the statistics are random. Pick the size of the smallest crater that you can reliably measure from the photograph. Now divide the image into four quadrants, and count all the craters in each quadrant. Call these numbers NA, NB, NC, and ND. They aren’t equal, right? However, the differences between them shouldn’t be much bigger than the counting errors, given by √ NA, √ NB, √ NC, and √ ND. Check that this is the case. In contrast, look at a similar size region of the lunar lava plains, the maria. This is an example of geological processes creating a young surface, where the smaller number of craters reflects a shorter cratering history.