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Useful Latex Equations used in R Markdown for Statistics
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--- | |
title: "Sample Equations used in Statistics" | |
output: html_document | |
--- | |
### Summations | |
### Without Indices | |
$\sum x_{i}$ | |
$\sum x_{i}^2$ | |
$\sum x_{i}y_{i}$ | |
#### With Indices - Inline Form | |
$\sum_{i=1}^n x_{i}$ | |
$\sum_{i=1}^n x_{i}^2$ | |
$\sum_{i=1}^n x_{i}y_{i}$ | |
#### With Indices - Display Form | |
$$\sum_{i=1}^n x_{i}y_{i}$$ | |
### Independent Samples | |
$$\mu_{\bar{x_{1}} - \bar{x_{2}}} = \mu_{1} - \mu_{2}$$ | |
$$\sigma_{\bar{x_{1}} - \bar{x_{2}}}^2 = \frac {\sigma_{1}^2}{n_{1}} + \frac{\sigma_{2}^2}{n_{2}}$$ | |
$$\mu_{\hat{p}_{1} - \hat{p}_{2}} = p_{1} - p_{2}$$ | |
$$\sigma_{\hat{p}_{1} - \hat{p}_{2}}^2 = \frac {p_{1}(1 - p_{1})}{n_{1}} + \frac {p_{2}(1 - p_{2})}{n_{2}}$$ | |
### Pooled Sample Variance | |
$$s_{p}^2 = \frac {(n_{1} - 1)s_{1}^2 + (n_{2} - 1)s_{2}^2}{n_{1} + n_{2} - 2}$$ | |
### Pooled Sample Proportion | |
$$\hat{p} = \frac {n_{1}\hat{p}_1 + n_{2}\hat{p}_{2}}{n_{1} + n_{2}}$$ | |
### Chi-Square Test | |
$$\chi^2 = \sum \frac {(O - E)^2}{E}$$ | |
### Correlations | |
$${SS}_{xx} = \sum (x - \bar{x})^2 = \sum x^2 - \frac {(\sum x)^2}{n}$$ | |
$${SS}_{xy} = \sum (x - \bar{x})(y - \bar{y}) = \sum xy - \frac {(\sum x)(\sum y)}{n}$$ | |
$$r = \frac {{SS}_{xy}}{\sqrt {{SS}_{xx}{SS}_{yy}}}$$ | |
### Regression | |
#### Population Regression Line | |
$$E(y) = \alpha + \beta{x}$$ | |
$$var(y) = \sigma^2$$ | |
#### Least Squares Line | |
$$\hat{y} = a + bx$$ | |
where | |
$$b = \frac {{SS}_{xy}}{{SS}_{xx}}$$ | |
and | |
$$\bar{y} = a + b\bar{x}$$ | |
#### Residual Sum of Squares | |
$$SSResid = \sum (y - \hat{y})^2 = \sum y^2 - a\sum y - b \sum xy$$ | |
#### Standard Errors | |
$$s_{e} = \sqrt \frac {SSResid}{n - 2}$$ | |
$$s_{b} = \frac {s_{e}}{\sqrt {{SS}_{xx}}}$$ | |
$$s_{a + bx} = s_{e} \sqrt {1 + \frac {1}{n} + \frac {(x - \bar{x})^2}{{SS}_{xx}}}$$ | |
for prediction: | |
$$se(y - \hat{y}) = s_{e} \sqrt {1 + \frac {1}{n} + \frac {(x - \bar{x})^2}{{SS}_{xx}}}$$ | |
### Variance | |
$$SSTr = \frac {T_{1}^2}{n_{1}} + \frac {T_{2}^2}{n_{2}} + ... + \frac {T_{k}^2}{n_{k}} - \frac {T^2}{n}$$ | |
$$SSTo = x_{1}^2 + x_{2}^2 + ... + x_{k}^2 - \frac {T^2}{n}$$ | |
$$SSE = SSTo - SSTr$$ |
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