(Guess the number of 'heads' out of 10 coin tosses, and you can keep this) |
Let me take you through a thought-experiment to show you this.
"I'm going to toss a coin 10 times. And to make it interesting, let me put this pineapple (Australian $50 note) on the table and make you this offer:
"If you guess the exact number of heads in the next ten coin tosses that I make, I will give you the $50. If you don't guess it exactly, I get to keep my $50. Are you willing?"
"Yes, sure."
So what number of coin tosses will be
H is for Heads
Most people guess somewhere around five (5/10). This
I now proceed to toss the coin, I catch it, I look, I call out the result. Here's a series of 10 coin tosses that I prepared beforehand:
"Heads
In a real version of this 'thought-experiment' (that I conduct in
Okay, so let's break it down.
H is for human intuition
1. There's a statistic which measures the observed result. It is very simple in this case, it is the number of "Heads" that come up, so let's just call it the H-statistic. (H is for heads).
2. The distribution of this H-statistic is known in advance: the expected result is 50% or thereabouts. So our expectation, the distribution of the H-Statistic under the "null hypothesis" is as pictured to the right.
3. The observed result was 9 out of 10 heads. Based on our understanding of the distribution at #2 above, we can all agree this result (9/10) is possible, but pretty improbable assuming the coin, the tosses and the calls are fair (which is our expectation under the null hypothesis). In fact, according, to the distribution (see chart), the probability of exactly 9 heads is . 01. The probability of 9 or higher (10 heads) is . 01 + .001 = .011. The probability of an extreme value of the H-statistic, say nine or more or one or less (for a two-tailed test) is . 011 + .011 = .022.
4. The key question here is whether we take an "improbable" result (p=. 022 or less for instance) and interpret it a s a "surprising" result given the expectation, or interpret it as "unlikely in this case". The laughter at seven or eight heads attests that many think that getting 9 heads out 10 exceeds the "significance level". In other words, they are saying, "Sure, the result is possible, but I'm going to call 'Bullsh !'" or in a statistician's language "statistical significance." We reject that the result was by chance, and conclude that something else was going on.
Step 1: Every statistic is simply a measure of an observed result. So χ2 (
(The statistical bell
Step 2: The distribution of statistics is known. In the example, the distribution of the H-statistic is shown in the chart. Essentially, the most likely result is 50:50 with declining probabilities
With the distribution in place, we can now look up the probability of any observed result, e.g., 9 as we observed. You can do that here for the H-statistic: the probability of success (i.e., "Heads") on a single trial is
Step 3: We set our significance (or alpha) level. This is our credulity level. If what we observe (the statistic) becomes sufficiently improbable (typically set as p
As an aside, statistical packages like SPSS incorrectly label p-values as "
Step 4: We now compare the p-level for the observed statistic with our pre
(The alternative? Plan B)
Giving up the possibility that 9/10 is just a surprising result and concluding that rather, it is a 'suspicious' result is called "rejecting the null hypothesis", Now, what do you think explains the result? Most proffer explanations like "The coin has two heads," "You tossed the coin a particular way,", some even claim "You're lying!" These are are all alternative hypotheses.
As it turns out, the statistician's significance level is probably a little stricter than the human's significance level (as based on the laughter measure). The probability of 9 or 10 heads or 0 or 1 heads (for the two-tailed test) is
H is for Happy?
So you see, statistical testing is not that difficult. (1) We start by calculating a statistic which is simply a measure of how far the observed is from what we might have expected (assuming no relationship). (2) Somewhere out there, some mathematical statisticians will have done the arcane calculations to produce a distribution of that statistic showing probabilities for each value from most probable (expected value) to the more distant and improbable values. To get the probability of our observed result, we simply look at the area under the curve for the statistic value, 'H' or more extreme. (3) We set our cut-point, our significance level, generally set at p =
Comments? Errors? (Yep, statistics
(Dr. Stephen P Holden is associate professor & executive director, 'Centre for Resurgent Enterprises', Jindal Global University)
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