Formulas

95% Confidence Interval: $s a m p l e s t a t i s t i c \pm 2 (s t a n d a r d e r r o r)$

Between Groups (Numerator) Degrees of Freedom: $d f_{b e t w e e n} = k - 1$; $k$ = number of groups

Binomial Random Variable Probability: $P (X = k) = (\binom{n}{k}) p^{k} (1 - p)^{n - k}$; $n$ = number of trials
$k$ = number of successes
$p$ = probability event of interest occurs on any one trial

Chi-Square ( $χ^{2}$ ) Test Statistic: $χ^{2} = Σ \frac{(O b s e r v e d - E x p e c t e d)^{2}}{E x p e c t e d}$

Complement of A: $P (A^{C}) = 1 - P (A)$

Conditional Probability of A Given B: $P (A ∣ B) = \frac{P (A \cap B)}{P (B)}$

Conditional Probability of B Given A: $P (B ∣ A) = \frac{P (A \cap B)}{P (A)}$

Confidence Interval for a Population Mean: $\overset{―}{x} \pm t^{*} \frac{s}{\sqrt{n}}$

Confidence Interval for the Difference Between Two Paired Means: ${\overset{―}{x}}_{d} \pm t^{*} (\frac{s_{d}}{\sqrt{n}})$; $t^{*}$ is the multiplier with $d f = n - 1$

Confidence Interval for the Difference Between Two Proportions: $({\hat{p}}_{1} - {\hat{p}}_{2}) \pm z^{*} \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}$

Confidence Interval for Two Independent Means: $({\bar{x}}_{1} - {\bar{x}}_{2}) \pm t^{*} \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}$

Confidence Interval of $p$ : Normal Approximation Method: $\hat{p} \pm z^{*} (\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}})$

Confidence Interval of $β_{1}$: $b_{1} \pm t^{*} (S E_{b_{1}})$; $b_{1}$ = sample slope
$t^{*}$ = value from the $t$ distribution with $d f = n - 2$
$S E_{b_{1}}$ = standard error of $b_{1}$

Degrees of Freedom: Chi-Square Test of Independence: $d f = (n u m b e r o f r o w s - 1) (n u m b e r o f c o l u m n s - 1)$

Estimated Degrees of Freedom: $d f = s m a l l e s t n - 1$

Expected Cell Value: $E = \frac{r o w t o t a l \times c o l u m n t o t a l}{n}$

Expected Count: $E x p e c t e d c o u n t = n (p_{i})$; $n$ is the total sample size
$p_{i}$ is the hypothesized proportion of the "ith" group

Finding Sample Size for Estimating a Population Proportion: $n = {(\frac{z^{*}}{M})}^{2} \tilde{p} (1 - \tilde{p})$; $M$ is the margin of error
$\tilde{p}$ is an estimated value of the proportion

Finding the Sample Size for Estimating a Population Mean: $n = \frac{z^{2} {\tilde{σ}}^{2}}{M^{2}} = {(\frac{z \tilde{σ}}{M})}^{2}$; $z$ = z multiplier for given confidence level
$\tilde{σ}$ = estimated population standard deviation
$M$ = margin of error

Five Number Summary: (Minimum, $Q_{1}$ , Median, $Q_{3}$ , Maximum)

General Form of 95% Confidence Interval: $s a m p l e s t a t i s t i c \pm 2 (s t a n d a r d e r r o r)$

General Form of a Test Statistic: $t e s t s t a t i s t i c = \frac{s a m p l e s t a t i s t i c - n u l l p a r a m e t e r}{s t a n d a r d e r r o r}$

Interquartile Range: $I Q R = Q_{3} - Q_{1}$

Intersection: $P (A \cap B) = P (A) \times P (B ∣ A)$

Mean of a Binomial Random Variable: $μ = n p$
Also known as $E (X)$

Observed Sample Mean Difference: ${\overset{―}{x}}_{d} = \frac{Σ x_{d}}{n}$; $x_{d}$ = observed difference

Odds

$o d d s = \frac{n u m b e r w i t h t h e o u t c o m e}{n u m b e r w i t h o u t t h e o u t c o m e}$

OR

$o d d s = \frac{r i s k}{1 - r i s k}$

Pearson's r: Conceptual Formula: $r = \frac{\sum z_{x} z_{y}}{n - 1}$
where $z_{x} = \frac{x - \overset{―}{x}}{s_{x}}$ and $z_{y} = \frac{y - \overset{―}{y}}{s_{y}}$

Pooled Estimate of $p$: $\hat{p} = \frac{{\hat{p}}_{1} n_{1} + {\hat{p}}_{2} n_{2}}{n_{1} + n_{2}}$

Population Mean: $μ = \frac{Σ x}{N}$

Power: $P o w e r = 1 - β$; $β$ = probability of committing a Type II Error.

Probability of Event A: $P (A) = \frac{N u m b e r i n g r o u p A}{T o t a l n u m b e r}$

Proportion: $P r o p o r t i o n = \frac{N u m b e r i n t h e c a t e g o r y}{T o t a l n u m b e r}$

Range: $R a n g e = M a x i m u m - M i n i m u m$

Relative Risk: $R e l a t i v e R i s k = \frac{R i s k i n G r o u p 1}{R i s k i n G r o u p 2}$

Residual: $e_{i} = y_{i} - {\hat{y}}_{i}$; $y_{i}$ = actual value of y for the ith observation
${\hat{y}}_{i}$ = predicted value of y for the ith observation

Risk: $R i s k = \frac{n u m b e r w i t h t h e o u t c o m e}{t o t a l n u m b e r o f o u t c o m e s}$

Sample Standard Deviation: $s = \sqrt{\frac{\sum (x - \overset{―}{x})^{2}}{n - 1}}$

Sample Variance: $s^{2} = \frac{\sum (x - \overset{―}{x})^{2}}{n - 1}$

Simple Linear Regression Line: Population: $\hat{y} = α + β x$

Simple Linear Regression Line: Sample: $\hat{y} = a + b x$; $\hat{y}$ = predicted value of $y$ for a given value of $x$
$a$ = $y$ -intercept
$b$ = slope

Slope: $b_{1} = r \frac{s_{y}}{s_{x}}$; $r$ = Pearson’s correlation coefficient between $x$ and $y$
$s_{y}$ = standard deviation of $y$
$s_{x}$ = standard deviation of $x$

Standard Deviation of a Binomial Random Variable: $σ = \sqrt{n p (1 - p)}$

Standard Deviation of the Differences: $s_{d} = \sqrt{\frac{\sum (x_{d} - {\overset{―}{x}}_{d})^{2}}{n - 1}}$

Standard Error: $\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}$

Sum of Squared Residuals: Also known as Sum of Squared Errors (SSE)
$S S E = \sum (y - \hat{y})^{2}$

Sum of Squares: $S S = \sum (x - \overset{―}{x})^{2}$

Test Statistic: $t e s t s t a t i s t i c = \frac{s a m p l e s t a t i s t i c - n u l l p a r a m e t e r}{s t a n d a r d e r r o r}$

Test Statistic for Dependent Means: $t = \frac{{\bar{x}}_{d} - μ_{0}}{\frac{s_{d}}{\sqrt{n}}}$; ${\overset{―}{x}}_{d}$ = observed sample mean difference
$μ_{0}$ = mean difference specified in the null hypothesis
$s_{d}$ = standard deviation of the differences
$n$ = sample size (i.e., number of unique individuals)

Test Statistic for Dependent Means: $t = \frac{{\bar{x}}_{d} - μ_{0}}{\frac{s_{d}}{\sqrt{n}}}$; ${\overset{―}{x}}_{d}$ = observed sample mean difference
$μ_{0}$ = mean difference specified in the null hypothesis
$s_{d}$ = standard deviation of the differences
$n$ = sample size (i.e., number of unique individuals)

Test Statistic for Two Independent Proportions: $z = \frac{{\hat{p}}_{1} - {\hat{p}}_{2}}{S E_{0}}$

Test statistic: One Group Proportion: $z = \frac{\hat{p} - p_{0}}{\sqrt{\frac{p_{0} (1 - p_{0})}{n}}}$; $\hat{p}$ = sample proportion
$p_{0}$ = hypothesize population proportion
$n$ = sample size

Union: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$

Within Groups (Denominator, Error) Degrees of Freedom

$d f_{w i t h i n} = n - k$

$n$ = total sample size with all groups combined

$k$ = number of groups

y-intercept: $b_{0} = \overset{―}{y} - b_{1} \overset{―}{x}$; $\overset{―}{y}$ = mean of $y$
$\overset{―}{x}$ = mean of $x$
$b_{1}$ = slope

z Test Statistic: One Group Mean: $z = \frac{\overset{―}{x} - μ_{0}}{\frac{σ}{\sqrt{n}}}$; $\overset{―}{x}$ = sample mean
$μ_{0}$ = hypothesized population mean
$s$ = sample standard deviation
$n$ = sample size

z-score: $z = \frac{x - \overset{―}{x}}{s}$; $x$ = original data value
$\overset{―}{x}$ = mean of the original distribution
$s$ = standard deviation of the original distribution