Some of you like my technical side notes ("wonkish" was the word you used, as I recall), so here's a crash course in the statistics that let particle systems look good. If you'd rather just focus on learning the "how" rather than the "why," you may want to give this article a miss -- the gentler Part 2 of this series is in the pipeline (it's already written, actually).

So as a side note to Part 1, we'll be using the same process by which we set the speed of the footballs (setting an average value and a measure of random variation) to accomplish a lot with particle systems (at least in Motion). You can control an awful lot about a given parameter just by setting two values: the average value of a parameter, and the amount of random variation you want in that parameter.

It might help if you have dealt with basic statistics: remember the "bell curve" of the normal (or Gaussian) distribution? It's called "normal" for a reason: an incredible range of natural phenomena fit the distribution. That's why it's useful when we're trying to fake natural phenomena.

So as a side note to Part 1, we'll be using the same process by which we set the speed of the footballs (setting an average value and a measure of random variation) to accomplish a lot with particle systems (at least in Motion). You can control an awful lot about a given parameter just by setting two values: the average value of a parameter, and the amount of random variation you want in that parameter.

It might help if you have dealt with basic statistics: remember the "bell curve" of the normal (or Gaussian) distribution? It's called "normal" for a reason: an incredible range of natural phenomena fit the distribution. That's why it's useful when we're trying to fake natural phenomena.

Importantly, the bell curve (like any statistical distribution) describes a large number (or

If this chart were strictly accurate (it's not), then we could figure that, if we were to pick a person at random from this population, that person would be very likely to have an IQ around 100, somewhat likely to have an IQ around 115, and very unlikely to have an IQ around 145. Now imagine that those numbers at the bottom described something other than IQ -- say, the speed at which your particles are emitted.

So one more time, things that are close to the middle hump of the bell curve happen pretty often, while things that are way out in the side tails of the curve don't happen very often at all. And this bell curve is what the computer uses as a "cheat sheet" to figure out what values to give individual particles so that, when you look at them all at once, they look like a normal group of things (population).

To describe a normal population with a bell curve, you don't need to know much about that population. The graph is going to take the shape of a bell, whatever you do. The vertical axis describes an imaginary probability value, so it'll always be the same. So all we need to know is what the horizontal axis -- the scale of "values" -- should look like for the population we're describing. And we can describe that with just two numbers: the average value, which sits under the middle hump of the curve; and the standard deviation, which is just a fancy way of saying "how spread out it is." For example, in the IQ chart above, each hatch mark on the horizontal axis is 15 points different from the neighboring hatch marks. Imagine that we were to change that to 5 points' difference, while keeping the bell shape the same. Now, the values look like this:

85 90 95

What we've done here is reduce the standard deviation, which is just a way of saying that we've reduced the amount of random variation around the average value that we set. If we were in Motion, and we had a pair of controls in our Particle Inspector called "IQ" and "IQ Random," it would be like we cranked down the "IQ Random" slider. Remember how we said that, in the original distribution, someone would be "somewhat likely" to have an IQ around 115? Well, now that we've reduced the amount of random variation, someone would be very unlikely to have an IQ around 115. But in both cases, the distribution of IQs is described by the normal distribution: as you get farther away from the average value, the likelihood of finding something with that value "fades out" gracefully.

Those two numbers describing the horizontal axis are basically what you're setting in your particle system controls: an average value, and a measure of how much you want your population of particles to spread out from that average value. And the reason that particle systems naturally look relatively believable with little effort is that, by and large, they rely on population distributions like the normal distribution to do their work.

*population*) of individual things -- be they people, or footballs, or drops of rain. The horizontal dimension of the curve describes a range of values. These "values" on the horizontal dimension could be anything -- people's height, footballs' speed, raindrops' volume. The vertical dimension describes the probability that, if I were to pick an individual thing at random, the individual thing would have that value -- the higher up on the curve I am, the higher the likelihood that I'll find that my individual thing has the value at that point. Unfortunately, I'm away from Photoshop right now, and all the egoistic bloggers seem to use the same example, so here's the only example I can find:If this chart were strictly accurate (it's not), then we could figure that, if we were to pick a person at random from this population, that person would be very likely to have an IQ around 100, somewhat likely to have an IQ around 115, and very unlikely to have an IQ around 145. Now imagine that those numbers at the bottom described something other than IQ -- say, the speed at which your particles are emitted.

So one more time, things that are close to the middle hump of the bell curve happen pretty often, while things that are way out in the side tails of the curve don't happen very often at all. And this bell curve is what the computer uses as a "cheat sheet" to figure out what values to give individual particles so that, when you look at them all at once, they look like a normal group of things (population).

To describe a normal population with a bell curve, you don't need to know much about that population. The graph is going to take the shape of a bell, whatever you do. The vertical axis describes an imaginary probability value, so it'll always be the same. So all we need to know is what the horizontal axis -- the scale of "values" -- should look like for the population we're describing. And we can describe that with just two numbers: the average value, which sits under the middle hump of the curve; and the standard deviation, which is just a fancy way of saying "how spread out it is." For example, in the IQ chart above, each hatch mark on the horizontal axis is 15 points different from the neighboring hatch marks. Imagine that we were to change that to 5 points' difference, while keeping the bell shape the same. Now, the values look like this:

85 90 95

**100**105 110 115What we've done here is reduce the standard deviation, which is just a way of saying that we've reduced the amount of random variation around the average value that we set. If we were in Motion, and we had a pair of controls in our Particle Inspector called "IQ" and "IQ Random," it would be like we cranked down the "IQ Random" slider. Remember how we said that, in the original distribution, someone would be "somewhat likely" to have an IQ around 115? Well, now that we've reduced the amount of random variation, someone would be very unlikely to have an IQ around 115. But in both cases, the distribution of IQs is described by the normal distribution: as you get farther away from the average value, the likelihood of finding something with that value "fades out" gracefully.

Those two numbers describing the horizontal axis are basically what you're setting in your particle system controls: an average value, and a measure of how much you want your population of particles to spread out from that average value. And the reason that particle systems naturally look relatively believable with little effort is that, by and large, they rely on population distributions like the normal distribution to do their work.

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