13.2 Collaborative statistics: formulas

Collaborative statistics using Page 1 / 1

This module provides an overview of Statistics Formulas used as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Binomial distribution

X ~ B (n, p)

P (X = x) = (\binom{n}{x}) p^{x} q^{n - x}

, for

x = 0, 1, 2, . . ., n

Binomial probability

P (x) = \frac{n!}{(n - x)! x!} \cdot p^{x} \cdot q^{n - x}

Central limit theorem

μ_{x} = μ

Central limit theorem (standard error)

σ_{x -} = \frac{σ}{\sqrt{n}}

Chi-square distribution

X ~ Χ_{df}^{2}

f (x) = \frac{x^{\frac{n - 2}{2}} e^{\frac{- x}{2}}}{2^{\frac{n}{2}} Γ (\frac{n}{2})}

x > 0

n

= positive integer and degrees of freedom

Confidence intervals (one population)

\hat{p} - E < p < \hat{p} + E

where

E = z_{\frac{a}{2}} \sqrt{\frac{p q}{n}}

x - E < μ < x + E

where

E = z_{\frac{a}{2}} \frac{σ}{\sqrt{n}}

t_{\frac{a}{2}} \frac{s}{\sqrt{n}}

Confidence interval (two populations)

({\hat{p}}_{1} - {\hat{p}}_{2}) - E < ({\hat{p}}_{1} - {\hat{p}}_{2}) < ({\hat{p}}_{1} - {\hat{p}}_{2}) + E

where

E = z_{\frac{a}{2}} \sqrt{\frac{{\hat{p}}_{1} {\hat{q}}_{1}}{n_{1}} + \frac{{\hat{p}}_{2} {\hat{q}}_{2}}{n_{2}}}

(x_{1} - x_{2}) - E < (μ_{1} - μ_{2}) < (x_{1} - x_{2}) + E

where

E = z_{\frac{a}{2}} \sqrt{\frac{σ_{1}}{n_{1}} + \frac{σ_{2}}{n_{2}}}

E = t_{\frac{a}{2}} \sqrt{\frac{{s_{1}}^{2}}{n_{1}} + \frac{{s_{2}}^{2}}{n_{2}}}

Goodness of fit

Chi-Square

= χ = \sum \frac{{(O - E)}^{2}}{E}

where

O =

observed and

E = \frac{(row total) (column total)}{(grand total)}

and

df = k - 1

Margin of error (mean, σ known)

E = z_{\frac{a}{2}} \frac{σ}{\sqrt{n}}

Margin of error (mean, σ unknown)

E = t_{\frac{a}{2}} \frac{s}{\sqrt{n}}

Margin of error (proportion)

E = z_{\frac{a}{2}} \sqrt{\frac{p q}{n}}

Mean

x = \frac{\sum x}{n}

Mean (binomial)

μ = n \cdot p

Mean (frequency table)

x = \frac{\sum f \cdot x}{\sum f}

Mean (probability distribution)

μ = \sum x \cdot P (x)

Normal distribution

X ~ N (μ, σ^{2})

f (x) = \frac{1}{σ \sqrt{2 π}} e^{\frac{{- (x - μ)}^{2}}{{2 σ}^{2}}}

- \infty < x < \infty

Sample size determination (mean)

n = [\frac{z_{\frac{a}{2}} σ}{E}]^{2}

Sample size determination (proportion)

n = \frac{(z_{\frac{a}{2}})^{2} \cdot 0.25}{E^{2}}

Sample size determination (where p-hat and q-hat are known)

n = \frac{(z_{\frac{a}{2}})^{2} \cdot p q}{E^{2}}

Standard deviation

s = \sqrt{\frac{{\sum (x - x)}^{2}}{n - 1}}

Standard deviation (binomial)

σ = \sqrt{n \cdot p \cdot q}

Standard deviation (frequency table)

s = \sqrt{\frac{{n (\sum (f \cdot x^{2})) - (\sum (f \cdot x))}^{2}}{n (n - 1)}}

Standard deviation (probability distribution)

σ = \sqrt{\sum [x^{2} \cdot P (x)] - μ^{2}}

Standard deviation (shortcut)

\sqrt{\frac{{n (\sum x^{2}) - (\sum x)}^{2}}{n (n - 1)}}

Standard score

z = \frac{z - x}{s}

\frac{z - μ}{σ}

Student-t distribution

X ~ t_{df}

Test for independence or homogeneity

Chi-Square

= χ = \sum \frac{{(O - E)}^{2}}{E}

where

O =

observed and

E = \frac{(row total) (column total)}{(grand total)}

and

df = (r - 1) (c - 1)

Tests statistic (one sample, μ known)

z = \frac{x - μ}{σ / \sqrt{n}}

Test statistic (one sample, μ unknown)

t = \frac{x - μ}{s / \sqrt{n}}

where

df = n - 1

Test statistic (one sample, proportion)

z = \frac{\hat{p} - p}{\sqrt{\frac{p q}{n}}}

Test statistic (two samples, proportions)

z = \frac{({\hat{p}}_{1} - {\hat{p}}_{2}) - (p_{1} - p_{2})}{\sqrt{\frac{p q}{n_{1}} + \frac{p q}{n_{2}}}}

where

= \frac{x_{1} + x_{2}}{n_{1} + n_{2}}

Test statistic (two samples, two means, μ unknown)

t = \frac{(x_{1} - x_{2}) - (μ_{1} - μ_{2})}{\sqrt{\frac{s^{2}}{n_{1}} + \frac{s^{2}}{n_{2}}}}

where

df =

smaller of

n_{1} - 1

n_{2} - 1

Uniform distribution

X ~ U (a, b)

f (X) = \frac{1}{b - a}

a < x < b

Variance

s^{2}

Variance (binomial)

σ^{2} = n \cdot p \cdot q

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Source: OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23

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