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Facts are stubborn, but statistics are more pliable.
- Mark Twain

Statistics are easy to abuse, or simply get wrong. Know what you are doing: Statistics+science books

e.g. Statistics, 4th Edition

A more pop look at statistics: The Signal and the Noise: Why So Many Predictions Fail but Some Don't

Hover over the form labels to get more information on each input/output


Enter a list of numbers separated by commas or spaces


Enter the observation value, the population mean of the distribution and the population standard deviation. N is the sample size used to obtain the observation.

You can also use the mean and std dev from the left:

$$z = \frac {x - \mu}{\sigma}$$

Confidence Intervals

Interval within which a hypothesis is tenable

Typical values are 90%, 95%, and 99%.

\(x-z_{\alpha/2}(\sigma/\sqrt{N})\lt\mu\lt x+z_{\alpha/2}(\sigma/\sqrt{N})\)

Use mean, std, and N from above?

Unpaired T-test

Use with two independent samples, where means, standard deviations, and sample sizes (N) can be different

Sample 1

Sample 2

? Value given is for a two-tailed test. Generally, with p < 0.05 one can "reject the null hypothesis" and say that the means are not just different by chance.


"n choose k": how many combinations can be obtained when choosing \(k\) items from a collection of \(n\) items?

Combinations: \(C = \tbinom nk = \frac{n!}{k!(n-k)!} \)

Permutations without repetion: \(P_0 = \frac{n!}{(n-k)!} \)

Permutations with repetion: \(P_r = n^k\)


? Number of combinations, where order does not matter

? Permutations without replacement.

? Permutations with replacement.

Binomial distribution, the probability of getting exactly \(k\) successes in \(n\) trials with independent "yes/no" probability \(p\): $$\Pr(X = k) = \tbinom nk p^k(1-p)^{n-k}$$


Get percentile of normal distribution

E.g., percentile 5 corresponds to quantile of \(\alpha=0.05\) which gives \(z_{\alpha} = -1.64\)

Graph: standard normal distribution (mean = 0, sigma = 1) with calculated quantile shown as a vertical line.

Power Analysis

Calculate minimum sample size needed given several conditions:

(difference of one sample mean from a constant, or difference of means for two matched samples)

? One or two tailed observation. For two tailed, half the value of alpha is allotted to each side of the normal distribution. Use one-sided if the value can only go to one side of the mean (e.g., if the test statistic can only be positive).

? Alpha: Type-I error, the probability of rejecting a null hypothesis that is true. A.k.a. 1-specificity, a.k.a., "confidence of the test", a.k.a, level of significance

? Power:sensitivity, a.k.a, 1-β (β is a Type-II error, failing to reject a null hypothesis that is false.)

? Effect size: A measure of the strength of the phenomenon. The smallest effect one expects/hopes to see. Could be the difference in means over the standard deviation.

? For two equal size independent samples, one would need double this amount for each

Visualizing Gaussian distributions

Also known as normal distributions


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