More comments on statistics

Since my last post, I’ve had a couple of other thoughts on the deceiving nature of statistics. In the era of such scientific achievements, people are understandably inclined to view scientific study and statistical analysis as the most proper means of determining truth. However, properly understood, statistics are merely a means of making a “best guess” based on limited information, and not a magical method for determining actual patterns. So let’s examine the coin toss…

I’ve spoken with a couple people lately for whom it’s a dull surprise that, when flipping a coin, we shouldn’t expect the coin tosses to result in a heads, tails, heads, tails pattern. The reason they seemed to believe this is that, assuming a properly balanced coin, the chances of any given coin toss landing on heads is 50/50. So that means, in theory, if I toss a coin twice, chances are I’ll get 1 heads and 1 tails. Four times? 2 of each. Six times? 3 of each, and so on. These people therefore expect that, when you toss the coin twice, if it landed on heads the first time, it will likely land on tails the following toss, and so on.

As another example, a lot of people have this same sort of misunderstanding of statistics so as to think that a random mix of black and white pebbles should be likely to generate a roughly checkerboard pattern– black, white, black, white– when, in fact, a checkerboard pattern is very un-random (or to use a real word, ordered).

However, lets look at the instance of 6 a coin being tossed 6 times. What are the chances that this pattern will be maintained throughout the entire span of the six tosses? In other words, what are the chances of getting either a heads-tails-heads-tails-heads-tails or tails-heads-tails-heads-tails-heads? In fact, the chances are no better than getting 6 heads or six tails. For any number of tosses, the chances of getting an alternating pattern is equal to getting a straight/constant result.

The reason for this is that the probability of any exact pattern is exactly equal. Getting the pattern heads-heads-tails-heads is no more or less likely than the pattern heads-tails-heads-heads or heads-heads-heads-heads. The reason why 3 heads and 3 tails is the most likely result of 6 coin tosses is because of the 64 possible patterns, 3 heads and 3 tails is the most common sum, accounting for 20 of the 64 possible patterns.

Notice that, although 3 heads and 3 tails is the most likely result, it’s not an overwhelmingly likely result. if you had to bet on whether the result will be exactly 3 heads/3 tails, the smart money would be on “no”. As a noted, a 3 to 3 split only accounts for 20 of the 64 patterns, meaning there’s roughly a 30% chance of reaching that result. However, the combinations reaching a result of 2/4 either way (either 2 heads and 4 tails or 4 heads and 2 tails) is more likely than that. In other words, the most probable result is probably not going to happen.

Now, all of this (along with my other post) is to say that, though statistical analysis is an extremely useful tool, statistics is only a predictive tool in an essentially unpredictable world, and it’s easy to put too much stock in the power of numbers to reveal truth.