Standard Deviations From the Beaten Path

One of my long term goals is to be old one day. One of those tingling distant fears I have is that being old will impair my ability to solve mental problems. I practically live off good feelings from problem solving. So this article from completely outside my expertise caught my eye: Cognitive activity and the cognitive morbidity of Alzheimer disease.

See those numbers in parentheses? Those pretty much sum up the statistics for each test they did. I’ll ignore the first two for the moment because I don’t have faith in my own ability to interpret them correctly for you, but the last one is the infamous p-value. This value tells you how likely your data set is assuming the null hypothesis.

In their case, the null hypothesis appears to be that cognitive activity has no effect on the progression of dementia for individuals at all three stages. The first group, which had no cognitive impairment at the onset of the study, saw less degradation with more activity. The p-value of 0.003 means there is a 0.3% chance of this if the null hypothesis is true, so they conclude that cognitive activity helps.

The second group with “mild cognitive impairment” had a result with a p-value of 0.300, or a %30 chance that the result came from the null hypothesis. The text of the article reflects this lack of a meaningful result. Lastly, the group with Alzheimer Disease, the degradation of cognitive function decreased with increased activity with p<0.001, considered a strong result for rejecting the null hypothesis.

Which is an important point. Hypothesis testing like this must be carefully constructed. All a p-value tells you is how likely your data set is assuming your null hypothesis. Rejecting the null hypothesis does not necessarily imply the result being tested. My interpretation of the above article as a whole would be that they’ve found compelling evidence that cognitive activity affects the rate of progression of dementia at various stages. What those specific effects are, they can’t really say yet. That’s why they’ll probably design another study to home in on the details.  I could go on about journalists jumping the gun and exaggerating results to satisfy people’s craving for instant gratification, but I think that’s already been said.

I find it ironic that we are in serious risk of running out of the 2nd most abundant element in the universe. Read on. It’s a fairly realistic concern. Helium is very light, and is ultimately unbound to Earth gravitationally. So once it escapes into the atmosphere, helium will diffuse off into space. Space, being very large, is an extremely inconvenient place to recover an escaping gas.

I never got far enough learning German to tell you if there are different genders for “intuition” and “logic”, but that’s not what this is about, anyhow. Alex at A Most Curious Planet asks, “Is Science Sexist?” Chad at Uncertain Principles responds with “Huh?” Don’t worry he explains.

To me this is a question that is only slightly more sane than the faux-feminist accusation that science is a man’s way of knowing that suppresses an equal, but unexplainable (and actually fictitious) woman’s way of knowing. In terms of answering practical scientific questions, science is the way of knowing. As Chad points out, intuition isn’t separate from science, which is not strictly logic. (more…)

On this day, July 20th, in the year 1969 humans landed on the moon. This is, in my opinion the coolest thing that has ever happened. Before graduating to the level of adult nerd, as an adolescent nerd I read everything I could about space. There is nothing uncool about space. Cosmic rays come from space, and that has a lot to do with why I’m so excited to be studying them, but the most awesome thing about space is that we landed humans on the moon 41 years ago.

There were no computers, not by modern standards anyway. There were no robots. There was just pure, unadulterated human chutzpah and a whole bunch of really smart people who found a way to make it happen. That’s why I wanted to do science. I wanted to figure out how to make things happen. And I have. Duck tape and cable ties. So take some time today to reflect on how awesome space is. Here’s an article in wired with less snark and more pretty pictures.

Remember those circular metal pieces you used to do your laundry back in college or graduate school? They had easily distinguished sides so that you could toss them in the air and let them land in one of two configurations, known for reasons now increasingly obscure due to a face lift on the American penny as “heads” or “tails”. Naturally, there is a   chance of either outcome–or as I mentioned earlier, over the course of an infinite number of measurements you’ll get an even 50/50 split.

What I want to do next, is add another coin. Each coin can be heads or tails so we have a total of four possible configurations, HH, TT, HT, or TH, with probability of each. You can continue to add more coins, resulting in an ever increasing number of unique outcomes, increasing like for coins tossed. Let’s say, though, that you don’t really care in which order the events occur, only how many coins land heads up. In this case, you’re concerned with the number of possible combinations, as opposed to the number of permutations. We can find the number of combinations by first finding the number of permutations and dividing this by the degeneracy.

So if we’re interested in tossing coins and getting heads, we start with a choice of coins for our first flip, then for our second and so on until we have coins for the final of our heads. While there might be some initial despair, this can be easily summed up with the factorial as . As I mentioned, if we don’t care in what order the coins were flipped, we need to divide by the degeneracy. For heads, there are possible orderings, so the number of combinations of items sampled ax a time is given by , which is frequently called “n choose x”.

Probability: We can get probability from this quite simply. We know the probability of an individual event occurring, 1/2 in this case, and we know how many ways in which it can happen, so we combine these to get , where . The name “binomial distribution” arises from the binomial theorem, which tells us that the sum over all possible values of for must be one.

Mean: I’ll spare the mathematical justification for the mean and standard deviation, as these are not straightforward. We can, however, get these correctly with a bit of intuition. In the case of the coin tossing experiment in which , we expect to get half heads and half tails. So for trials, we expect of our outcome of interest leading to .

Standard Deviation: The intuitive tack to this one is a little less, well, intuitive. We can exploit a trick called “expectation of the square minus square of the expectation” to get the variance, . For a single coin flip, the mean of the square is just just . Conveniently, the square of the mean is . So we can calculate . For trials, it’s a simple sum so we can get the standard deviation for the binomial distribution .

Naturally, we don’t need a coin flip. It could work just as well for rolling dice where the probability of a particular outcome on one die is 1/6. If you’re sufficiently nerdy to have ever played the game Settlers of Catan, you might recall the dots under each number on the game board. These represent combinations on two dice that will yield that result. So really, any game in which you roll dice can be partially characterized by the binomial distribution. Next, we can take the limit as the probability becomes very small and arrive at the Poisson distribution, which is extremely useful for understanding the results of surveys and polls, so stay tuned for that.

On some level, statistics is the process of describing distributions, which help us describe the probability of an event given a certain set of circumstances. I’ll get into the particulars of a few different types of statistical distributions like the binomial, Poisson, and Gaussian distributions which are commonly used to describe scientific data. Before we get into that, though, I wanted to just talk a bit about the statistical distribution as a concept,  particularly the idea of a parent distribution and a sample distribution.

A lot of physics boils down to trying to characterize a random processes. The ubiquitous, quintessential example is a coin flip. If you want to test the fairness of a coin you might flip it many times and count the number of heads.  What you would have collected is a sample distribution from the parent distribution of the coin describing the randomness of the coin. If you attempt this experiment with a real coin, you will likely get something very close to the canonical distribution for a coin flip, the binomial distribution, because for a probability close to 0.5 the sample tends to lie very close to the parent.

For other distributions the important distinction becomes much more apparent. I threw together a little python script you can play with yourself that generates 100 numbers from a randomly seeded pseudo-random number generator. It’ll then plot these data on the same axis as the probability density function of the parent distribution. Potential results might look like this. The values of the mean and standard deviation for a Gaussian distribution are listed in the legend, as well as the calculated values for a sample of 100 “measurements”. A common convention is to refer to the statistics of the parent distribution by the Greek letters and and the sample statistics by the roman letters and (or for the deviance).

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Bad Science writer Ben Goldacre wrote about how “Guns don’t kill people, puppies do,” where he points out some ways in which the probabilities of things can be a bit confusing. An extremely common example, which was messed up by a British newspaper, is the probability of multiple people (siblings in this case) sharing a birthday.

Like he says, the trick is that it doesn’t matter when the first child was born, only that the next two were born on the same day. This is a trivial case in which the binomial coefficient simply cancels out the leading 1/365. In general, though, you can use the binomial coefficient (sometimes called “n choose k”) to calculate probabilities where you can get the same result in multiple ways.

So if you’re guessing the suit of a hidden playing card, you have a 1/4 chance of guessing correctly. If you guess for 10 different cards and get 5 correct, you might be tempted to say your feat was as likely as , and that you might have psychic powers. What you’re missing, though, is the number of ways you could have guessed correctly, , which means the probability of correctly guessing half the cards was 0.058. You could still be psychic, but now it doesn’t seem quite as likely.

Tune in later for a crash-course introduction to P-values and hypothesis testing, so we can conclude with more certainty one way or the other on your psychic abilities.

Two Wi-Fi (what does that even stand for, anyway?) stories from recently.

1. A device that claims to get power from Wi-Fi.
2. A man who claims his neighbor’s Wi-Fi signal makes him sick.

I doubt the plausibility of both of these claims. A run-of-the-mill household wireless router is rated by the manufacturer as having a total power output of 15 dBmW. This is a logarithmic scale describing the power output of radiative devices in such a way that the same units can be used for very powerful and not so powerful antennas. In good old SI units, this is about 32 mW.

Electromagnetic waves typically propagate isotropically. That is, wavefronts take the form of spheres expanding in every direction from the point of origin at the speed of light. When we talk about the power of a transmitter we’re usually talking about the power obtained by integrating over the size of the wavefront. So the power is the same from any point that is the same distance from the antenna, but constantly decreasing with distance. Because the wavefront is spherical, the rate at which the power density falls off is one over the square of the distance.

An order of magnitude estimate for the power of an FM radio station is 100 kW. How  far away would we have to be from the radio station in order to see the same power density (watts per meter squared) as we would 1m away from the Wi-Fi router? About 1.8 km (1.1 miles). So I might ask, does being near a radio station make the electromagnetically sensitive gentleman ill? One mile isn’t an unreasonable distance to be from a transmitting tower. And if it’s possible to pick up power from a 32 mW signal, why hasn’t it been developed for FM radio earlier? And what’s the efficiency? Is 32 mW (assuming you’re right next to the transmitter) enough to charge a battery, anyway?

CDMS stands for Cold Dark Matter Search. The experiments objectives are to find evidence for the existence of dark matter–matter that we have so far only been able to see due to its gravitational interaction–that is low in kinetic energy, as opposed to “hot” dark matter like neutrinos. The most compelling form of dark matter are Weakly Interacting Massive Particles, or WIMPs. They’ve just released their 2007-2008 results, which you can find here, straight from the source: http://cdms.berkeley.edu/

So what have they seen? One strong point of CDMS is a very intense understanding of their background, that is, events that look like they could be coming from WIMPs, but probably aren’t for a variety of reasons. So after all is said and done and they’ve isolated the WIMP signal they are left with two events. But this still doesn’t mean they’ve discovered dark matter! In their arXiv paper they go on to say that there is a 23% probability of background creating two or more events in the region that would get past their filters. So there’s about one out of four chances that it was missed background, so as they say, “These expectations indicate that the result of this analysis cannot be interpreted as significant evidence for WIMP interactions, but we cannot reject either event as signal.”

And hopefully your excitement is not oscillating too rapidly, but in case your disappointed by the news that no especially compelling evidence for WIMP interactions has been found, they will be able to set some awesome upper limits on the WIMP-nucleon cross-section.

Are you in a warm, dry environment? If so, go open your refrigerator door, stand there for a few seconds and get something out if you’d like. Wait a short while, another few seconds or so. Maybe long enough to pour a glass of water if you’re the kind of person who keeps that in your refrigerator. Now try to open the door to your refrigerator again. Feel that? It’s hard, or at least harder to open!

For those of you who are either not living in a desert or pretty much anywhere in the northern hemisphere at the moment this will probably not do much of anything, so  I guess you’ll just have to take my word for it. But what you would be experiencing is the creation of a pressure difference between your kitchen and your refrigerator. The why isn’t particularly hard to understand, although it is a little unintuitive.

So, let’s start with a refrigerator, shall we?

The air inside is cold(er) and the air outside is warm(er), and dry. After opening the door, warm air is introduced to the interior, mostly by way of the closing door itself. Once the door is closed, the warm dry air cools rapidly. Cool air takes up less space than warm air (the same reason hot air balloons can fly), so what you wind up with is a rather surprising pressure difference between the inside of the refrigerator and the rest of the kitchen.

You might be wondering. Why dry air. Water takes up way less space than water vapor, so wouldn’t you get a stronger vacuum that way? Well, yes and no. If you had a really tight seal on your ‘fridge that could work. The problem is that the phase transition has an overhead so you’d need a bigger temperature difference to achieve the same effect and is also slower. The dry air is so effective because it can cool down and “shrink” rapidly compared to the rate at which pressure is exchanged  by the seal on the door.

This was an incredible source of amusement while I was in Argentina for a couple of weeks. Someone would open the refrigerator to extract the bottle of water, say, pour a glass, then walk back with the bottle in hand to return it. Except the pressure difference was so great, it took two hands to open the door. An entire evening was passed trying to explain this physically, so I figured I had to share.