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❶The chi-square test takes into account neither the number of experiments performed nor the probability distribution of the expected outcomes; it is valid only as the number of experiments becomes large, resulting in substantial numbers for the most probable results.
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The latter document is more than K bytes and will take a while to download, and contains a very large table which some Web browsers, particularly on machines with limited memory, may not display properly. The normal distribution gives the probability for x heads in n flips as:. To show how closely the probability chart approaches the normal distribution even for a relatively small number of flips, here's the normal distribution plotted in red, with the actual probabilities for number of heads in flips shown as blue bars.

The probability the outcome of an experiment with a sufficiently large number of trials is due to chance can be calculated directly from the result, and the mean and standard deviation for the number of trials in the experiment. For additional details, including an interactive probability calculator, please visit the z Score Probability Calculator.

This is all very persuasive, you might say, and the formulas are suitably intimidating, but does the real world actually behave this way? Well, as a matter of fact, it does, as we can see from a simple experiment.

Get a coin, flip it 32 times, and write down the number of times heads came up. Now repeat the experiment fifty thousand times. When you're done, make a graph of the number of flip sets which resulted in a given number of heads.

Hmmmm…32 times 50, is 1. Instead of marathon coin-flipping, let's use the same HotBits hardware random number generator our experiments employ.

It's a simple matter of programming to withdraw 1. The results from this experiment are presented in the following graph. The red curve is the number of runs expected to result in each value of heads, which is simply the probability of that number of heads multiplied by the total number of experimental runs, 50, The blue diamonds are the actual number of 32 bit sets observed to contain each number of one bits.

It is evident that the experimental results closely match the expectation from probability. Just as the probability curve approaches the normal distribution for large numbers of runs, experimental results from a truly random source will inexorably converge on the predictions of probability as the number of runs increases. If your Web browser supports Java applets, our Probability Pipe Organ lets you run interactive experiments which demonstrate how the results from random data approach the normal curve expectation as the number of experiments grows large.

Performing an experiment amounts to asking the Universe a question. For the answer, the experimental results, to be of any use, you have to be absolutely sure you've phrased the question correctly. When searching for elusive effects among a sea of random events by statistical means, whether in particle physics or parapsychology, one must take care to apply statistics properly to the events being studied.

Misinterpreting genuine experimental results yields errors just as serious as those due to faults in the design of the experiment. Evidence for the existence of a phenomenon must be significant , persistent , and consistent. Statistical analysis can never entirely rule out the possibility that the results of an experiment were entirely due to chance—it can only calculate the probability of occurrence by chance.

Only as more and more experiments are performed, which reproduce the supposed effect and, by doing so, further decrease the probability of chance, does the evidence for the effect become persuasive. To show how essential it is to ask the right question, consider an experiment in which the subject attempts to influence a device which generates random digits from 0 to 9 so that more nines are generated than expected by chance.

Each experiment involves generation of one thousand random digits. We run the first experiment and get the following result:.

There's no obvious evidence for a significant excess of nines here we'll see how to calculate this numerically before long. There was an excess of nines over the chance expectation, , but greater excesses occurred for the digits 3 , 5 , 6 , and 7.

But take a look at the first line of the results! What's the probability of that happening? Just the number of possible numbers of d digits which contain one or more sequences of p or more consecutive nines:.

So then, are the digits not random, after all? Might our subject, while failing to influence the outcome of the experiment in the way we've requested, have somehow marked the results with a signature of a thousand-to-one probability of appearing by chance? Or have we simply asked the wrong question and gotten a perfectly accurate answer that doesn't mean what we think it does at first glance?

The latter turns out to be the case. Note the order in which we did things. We ran the experiment, examined the data, found something seemingly odd in it, then calculated the probability of that particular oddity appearing by chance. That alone reduces the probability of occurrence by chance to one in ten. It is, in fact, very likely you'll find some pattern you consider striking in a random digit number.

But, of course, if you don't examine the data from an experiment, how are you going to notice if there's something odd about it? Now we'll see how a hypothesis is framed, tested by a series of experiments, and confirmed or rejected by statistical analysis of the results.

So, let's pursue this a bit further, exploring how we frame a hypothesis based on an observation, run experiments to test it, and then analyse the results to determine whether they confirm or deny the hypothesis, and to what degree of certainty.

Based on this observation we then suggest:. We can now proceed to test this experimentally. To be correct, it's important to test each digit sequence separately, then sum the results for consecutive sequences.

We will perform, then, the following experiment. Every sequences, we'll record the number of occurrences, repeating the process until we've generated a thousand runs of a million digits—10 9 digits in all. The number of occurrences expected by chance, 0. At the outset, the results diverged substantially from chance, as is frequently the case for small sample sizes. But as the number of experiments increased, the results converged toward the chance expectation, ending up in a decreasing magnitude random walk around it.

So far, we've seen how the laws of probability predict the outcome of large numbers of experiments involving random data, how to calculate the probability of a given experimental result being due to chance, and how one goes about framing a hypothesis, then designing and running a series of experiments to test it. Now it's time to examine how to analyse the results from the experiments to determine whether they provide evidence for the hypothesis and, if so, how much. Applicable to any experiment where discrete results can be measured, it is used in almost every field of science.

The chi-square test is the final step in a process which usually proceeds as follows. No experiment or series of experiments can ever prove a hypothesis; one can only rule out other hypotheses and provide evidence that assuming the truth of the hypothesis better explains the experimental results than discarding it. In many fields of science, the task of estimating the null-hypothesis results can be formidable, and can lead to prolonged and intricate arguments about the assumptions involved.

Experiments must be carefully designed to exclude selection effects which might bias the data. Fortunately, retropsychokinesis experiments have an easily stated and readily calculated null hypothesis: Anybody can score better than chance at coin flipping if they're allowed to throw away experiments that come out poorly! Finally, the availability of all the programs in source code form and the ability of others to repeat the experiments on their own premises will allow independent confirmation of the results obtained here.

So, as the final step in analysing the results of a collection of n experiments, each with k possible outcomes, we apply the chi-square test to compare the actual results with the results expected by chance, which are just, for each outcome, its probability times the number of experiments n. Mathematically, the chi-square statistic for an experiment with k possible outcomes, performed n times, in which Y 1 , Y 2 ,… Y k are the number of experiments which resulted in each possible outcome, where the probabilities of each outcome are p 1 , p 2 ,… p k is:.

It's evident from examining this equation that the closer the measured values are to those expected, the lower the chi-square sum will be. Unfortunately, there is no closed form solution for Q , so it must be evaluated numerically.

If your Web browser supports JavaScript, you can use the Chi-Square Calculator to calculate the probability from a chi-square value and number of possible outcomes, or calculate the chi-square from the probability and the number of possible outcomes.

In applying the chi-square test, it's essential to understand that only very small probabilities of the null hypothesis are significant. The chi-square test takes into account neither the number of experiments performed nor the probability distribution of the expected outcomes; it is valid only as the number of experiments becomes large, resulting in substantial numbers for the most probable results. If a hypothesis is valid, the chi-square probability should converge on a small value as more and more experiments are run.

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