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# Statistics Calculators

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:

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

### 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

### 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: $$P_r = n^k$$

choose

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

#### Disclaimer:

While we have done our best to assure accurate results, the authors of this website do not make any representation or warranty, express or implied, regarding the calculators on this website, nor assume any liability for its use. The code implementation is the intellectual property of the developers.

Please let the webmaster know if you find any errors or discrepancies.
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