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

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**

### Mean/Median/Std

Enter a list of numbers separated by commas or spaces

### Z-score

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:

### Confidence Intervals

Interval within which a hypothesis is tenable

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

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

### Unpaired T-test

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

### Combinatorics

"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: \(P_r = n^k\)

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

### Percentile

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)

### Visualizing Gaussian distributions

Also known as normal distributions