Formula Sheet

This is the course formula sheet, organized by topic in the order the book teaches them. Every symbol is defined the first time it appears, and the notation matches the glossary and the mosaic/BSDA functions you run in the labs. This page is reference-sheet material — the kind your instructor may allow on an exam (see the course randomization & exam policy).

11. Descriptive statistics¶

For a numerical variable with values $x_1, x_2, \dots, x_n$ ( $n$ = number of observations):

Quantity	Formula	Symbols
Sample mean	$\displaystyle \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$	$\bar{x}$ = sample mean; $x_i$ = the $i$ -th value; $\sum$ = “sum over all values”
Sample variance	$\displaystyle s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2$	$s^2$ = sample variance; $n-1$ = degrees of freedom
Sample standard deviation	$s = \sqrt{s^2}$	$s$ = typical distance of a value from $\bar{x}$
Range	$\text{max} - \text{min}$	largest minus smallest value
Interquartile range	$\text{IQR} = Q_3 - Q_1$	$Q_1, Q_3$ = first and third quartiles (25th, 75th percentiles)
Outlier rule (1.5 × IQR)	below $Q_1 - 1.5\,\text{IQR}$ or above $Q_3 + 1.5\,\text{IQR}$	flags unusually distant points
z-score (standardized value)	$\displaystyle z = \frac{x - \bar{x}}{s}$	$z$ = number of SDs $x$ lies from the mean

Population versions replace $\bar{x}$ with $\mu$ (population mean), $s$ with $\sigma$ (population SD), and divide the variance by $N$ (population size) instead of $n-1$ .

22. Probability¶

Rule	Formula	Symbols
Probability bounds	$0 \le P(A) \le 1$	$P(A)$ = probability of event $A$
Complement	$P(A^c) = 1 - P(A)$	$A^c$ = “ $A$ does not happen”
Addition (general)	$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$	subtract the overlap once
Addition (disjoint)	$P(A \text{ or } B) = P(A) + P(B)$	disjoint = cannot both happen
Multiplication (independent)	$P(A \text{ and } B) = P(A)\,P(B)$	only when $A, B$ independent
Conditional probability	$\displaystyle P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}$	$P(A\mid B)$ = probability of $A$ given $B$ occurred
Independence check	$A,B$ independent iff $P(A\mid B) = P(A)$

33. Random variables & the Normal model¶

For a discrete random variable $X$ taking values $x_i$ with probabilities $P(x_i)$ :

Quantity	Formula	Symbols
Expected value (mean)	$\displaystyle \mu = E(X) = \sum_i x_i\,P(x_i)$	$\mu$ = long-run average of $X$
Variance	$\displaystyle \sigma^2 = \sum_i (x_i - \mu)^2\,P(x_i)$	$\sigma^2$ = spread of $X$
Standard deviation	$\sigma = \sqrt{\sigma^2}$

Binomial model (count of successes in $n$ independent trials, each with success probability $p$ ):

P(X = k) = \binom{n}{k}\,p^k (1-p)^{\,n-k}, \qquad \mu = np, \qquad \sigma = \sqrt{np(1-p)}

where $k$ = number of successes and $\binom{n}{k} = \dfrac{n!}{k!\,(n-k)!}$ is the number of ways to choose $k$ successes.

Normal distribution $N(\mu, \sigma)$ — standardize any value to a $z$ -score:

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

68–95–99.7 rule: about 68%, 95%, and 99.7% of values fall within 1, 2, and 3 standard deviations of the mean. (Exact areas, computed from the Normal model: 0.6827, 0.9545, 0.9973 — see the distribution tables.)

44. Sampling distributions & the CLT¶

The sampling distribution of a statistic describes how it varies from sample to sample. Its standard deviation is the standard error (SE).

Statistic	Center	Standard error	Symbols
Sample mean $\bar{x}$	$\mu$	$\displaystyle SE_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \;\approx\; \frac{s}{\sqrt{n}}$	$\sigma$ = pop. SD; $s$ = sample SD; $n$ = sample size
Sample proportion $\hat{p}$	$p$	$\displaystyle SE_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$	$p$ = pop. proportion; $\hat{p}$ = sample proportion

Central Limit Theorem. When observations are independent and $n$ is large enough, the sampling distribution of $\bar{x}$ (or $\hat{p}$ ) is approximately $N\!\left(\mu,\ \sigma/\sqrt{n}\right)$ regardless of the population’s shape. The $\sqrt{n}$ in the denominator is why quadrupling the sample size only halves the standard error.

55. Confidence intervals (general form)¶

\text{estimate} \;\pm\; (\text{critical value}) \times (\text{standard error})

The critical value comes from the Normal ( $z^\star$ ) or $t$ ( $t^\star_{df}$ ) distribution and matches the confidence level; the second factor is the margin of error, $E$ .

Parameter	Interval	Critical value	Conditions
One proportion $p$	$\displaystyle \hat{p} \pm z^\star \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	$z^\star$	independence; $n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$
Diff. of proportions $p_1-p_2$	$\displaystyle (\hat{p}_1-\hat{p}_2) \pm z^\star \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$	$z^\star$	success–failure $\ge 10$ in both groups
One mean $\mu$	$\displaystyle \bar{x} \pm t^\star_{df}\,\frac{s}{\sqrt{n}}$ , $\;df = n-1$	$t^\star_{df}$	independence; roughly Normal or $n \ge 30$
Diff. of means $\mu_1-\mu_2$	$\displaystyle (\bar{x}_1-\bar{x}_2) \pm t^\star_{df}\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$	$t^\star_{df}$	independence; each group Normal or $n\ge 30$
Paired mean $\mu_d$	$\displaystyle \bar{x}_d \pm t^\star_{df}\,\frac{s_d}{\sqrt{n}}$ , $\;df = n-1$	$t^\star_{df}$	differences independent and roughly Normal

Common $z^\star$ values (two-sided), computed from the Normal model: $z^\star_{90\%} = 1.645$ , $z^\star_{95\%} = 1.960$ , $z^\star_{99\%} = 2.576$ .

66. Hypothesis tests — the general recipe¶

Every test follows the same arc. The test statistic measures how far the estimate sits from the null value, in standard-error units:

\text{test statistic} = \frac{\text{point estimate} - \text{null value}} {\text{standard error (under } H_0)}

A large statistic produces a small p-value = $P(\text{result at least this extreme} \mid H_0 \text{ true})$ . Reject $H_0$ when p-value $< \alpha$ (the significance level).

6.16.1 Proportions¶

One proportion ( $H_0: p = p_0$ ): the SE uses the null value $p_0$ .

z = \frac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}

Two proportions ( $H_0: p_1 = p_2$ ): use the pooled proportion $\hat{p} = \dfrac{x_1 + x_2}{n_1 + n_2}$ (total successes over total sample) for the test SE:

z = \frac{\hat{p}_1 - \hat{p}_2} {\sqrt{\hat{p}(1-\hat{p})\left(\dfrac{1}{n_1} + \dfrac{1}{n_2}\right)}}

where $x_1, x_2$ = successes and $n_1, n_2$ = sample sizes in each group.

6.26.2 Means (t-tests)¶

One mean ( $H_0: \mu = \mu_0$ ), $df = n-1$ :

t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}

Paired mean — reduce the pairs to differences $d_i$ , then run a one-sample $t$ on the differences ( $\bar{x}_d$ = mean difference, $s_d$ = SD of differences, $df = n-1$ ):

t = \frac{\bar{x}_d - 0}{s_d/\sqrt{n}}

Two independent means ( $H_0: \mu_1 = \mu_2$ ):

t = \frac{(\bar{x}_1 - \bar{x}_2) - 0} {\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}}

with degrees of freedom from software (Welch’s approximation; a conservative hand value is $df = \min(n_1 - 1,\ n_2 - 1)$ ).

6.36.3 Chi-square ( $\chi^2$ )¶

For a table of observed counts $O$ with expected counts $E$ computed under the null:

\chi^2 = \sum \frac{(O - E)^2}{E}

Test	Expected count $E$	Degrees of freedom
Goodness of fit ( $k$ categories)	$E_i = n\,p_i$	$df = k - 1$
Independence ( $r \times c$ table)	$E = \dfrac{(\text{row total})(\text{column total})}{\text{grand total}}$	$df = (r-1)(c-1)$

$O$ = observed count in a cell; $E$ = expected count if $H_0$ were true; $p_i$ = hypothesized proportion for category $i$ ; $r, c$ = numbers of rows and columns. Condition: every expected count $E \ge 5$ .

6.46.4 ANOVA (comparing many means)¶

$H_0$ : all group means equal. With $k$ groups and $N$ total observations, the $F$ -statistic is the ratio of between-group to within-group variation:

F = \frac{\text{MSG}}{\text{MSE}} = \frac{\text{SSG}/(k-1)}{\text{SSE}/(N-k)}

Term	Formula	Meaning
SSG (between)	$\displaystyle \sum_{j=1}^{k} n_j (\bar{x}_j - \bar{x})^2$	how far group means spread from the overall mean
SSE (within)	$\displaystyle \sum_{j=1}^{k}\sum_{i} (x_{ij} - \bar{x}_j)^2$	leftover variation inside the groups
MSG	$\text{SSG}/(k-1)$	between-group mean square; $df_1 = k-1$
MSE	$\text{SSE}/(N-k)$	within-group mean square; $df_2 = N-k$

$k$ = number of groups; $N$ = total sample size; $n_j$ = size of group $j$ ; $\bar{x}_j$ = mean of group $j$ ; $\bar{x}$ = grand mean; $x_{ij}$ = the $i$ -th value in group $j$ . A large $F$ (small p-value) signals that at least one group mean differs.

77. Simple linear regression (SLR)¶

Model: $\;\hat{y} = b_0 + b_1 x\;$ — the least-squares line through points $(x_i, y_i)$ .

Quantity	Formula	Symbols
Correlation	$\displaystyle r = \frac{1}{n-1}\sum \left(\frac{x_i - \bar{x}}{s_x}\right)\!\left(\frac{y_i - \bar{y}}{s_y}\right)$	$r \in [-1,1]$ ; $s_x, s_y$ = SDs of $x$ and $y$
Slope	$\displaystyle b_1 = r\,\frac{s_y}{s_x}$	change in $\hat{y}$ per 1-unit rise in $x$
Intercept	$b_0 = \bar{y} - b_1 \bar{x}$	$\hat{y}$ when $x = 0$
Predicted value	$\hat{y}_i = b_0 + b_1 x_i$	model’s estimate for observation $i$
Residual	$e_i = y_i - \hat{y}_i$	observed minus predicted
R-squared	$R^2 = r^2$	fraction of variation in $y$ explained by $x$
Slope test	$\displaystyle t = \frac{b_1 - 0}{SE_{b_1}}$ , $\;df = n-2$	tests $H_0: \beta_1 = 0$ (no linear relationship)
Slope CI	$b_1 \pm t^\star_{df}\,SE_{b_1}$	$\beta_1$ = true population slope

Here $\beta_0, \beta_1$ are the population intercept and slope; $b_0, b_1$ are their sample estimates; $SE_{b_1}$ = standard error of the slope (from software). Conditions: linearity, independent residuals, roughly Normal residuals, and constant residual spread (equal variance).