Overview

Notes from John Walker: https://www.fourmilab.ch/rpkp/experiments/statistics.html.

Definitions

• $\sigma$ = standard deviation
• $\sigma^2$ = variance

Permutations

The number of permutations of $n$ different things taken $n$ at a time is $n!$.
The number of permutations of $n$ different things taken $x$ at a time is $\binom{n}{x}$ and calculated as $\frac{n!}{x!(n-x)!}$ where ${n}$ and ${x}$ are positive integers.

Flipping a Coin

If a coin is flipped four times, there are 16 possible outcomes:

Plotting this data results in this bar graph:

Probability

Each flip has an equal chance of coming up heads or tails, so all 16 possible outcomes are equally probable.

However, since there are 6 ways to get 2 heads, the probability of two heads is greater than any other outcome:

Calculating Probability of One Outcome or Another

Sum the probabilities. The probability of getting either one or two heads $0.25 + 0.375 = 0.625$.

Calculating Probability of One Outcome and Another From Two Separate Experiments

Multiply the probabilities. The probability one head in each of two successive sets of four flips is $0.25 * 0.25 = 0.0625$

The probability of any number of heads $x$ in any number of flips $n$ is ${\binom{n}{x}\over2^n} = \frac{n!}{{2^n}x!(n-x)!}$

The normal distribution of probability equation

The probability for $x$ heads in $n$ flips where $\mu=n/2$
${{1}\over{\sqrt{2\pi\sigma}}}e^-{(x-\mu)^2\over{2\sigma^2}}$

Experiments and Evidence

• 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.
• 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 results than discarding it.

$\Chi^2$ (Chi-Square) Test

• The measure of the significance to which experimental results support or refute a hypothesis.
• It is applicable to any experiment where discrete results can be measured.
• If the probability that the null hypothesis can explain the experimental results is above 1%, an experiment is generally not considered evidence of a different hypothesis.

The chi-square test is the final step in the following process:

1. A phenomenon is suggested to exist based either on theory or examination of empirical data.
2. A hypothesis is framed incorporating the supposed phenomenon.
3. An experiment is designed to test the hypothesis.
   1. The experiment must produce results which differ based on the validity of the hypothesis.
   2. The results for the experiment for the null hypothesis (the hypothesis-false case) are calculated.
4. A series of experiments are conducted and the results tabulated.
5. The chi-square test is applied to determine the significance of the discrepancy between the calculated null hypothesis case and the experimental results, and the probability that the discrepancy is the result of chance.

The chi-square statistic for an experiment with $k$ possible values 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:
$\Chi^2=\sum_{\substack{1\le{s}\le{k}}} {(Y_s-np_s)^2\over{np_s}}$