Course Formula Sheet - Statistics in the Modern World (MATH 1209)

This sheet gathers every formula used in MATH 1209, organized by topic, with each symbol defined the first time it appears. The notation matches the weekly units: Roman letters (such as $\bar{x}$ , $s$ , $\hat{p}$ ) are statistics computed from a sample, and Greek letters (such as $\mu$ , $\sigma$ , $p$ ) are parameters that describe a whole population. Round only at the end of a calculation, never in the middle. A quick guide to entering these on a calculator is in the TI-83/84 Quick Guide.

1Summarizing data (Ch 2)¶

Quantity	Formula	Symbols
Sample mean	$\bar{x} = \dfrac{\sum x_i}{n}$	$x_i$ = each data value; $n$ = sample size; $\sum$ = “add up”
Sample variance	$s^2 = \dfrac{\sum (x_i - \bar{x})^2}{n-1}$	average squared distance from the mean
Sample standard deviation	$s = \sqrt{\dfrac{\sum (x_i - \bar{x})^2}{n-1}}$	$s$ = typical distance of a value from $\bar{x}$ , in original units
Range	$\text{max} - \text{min}$	largest value minus smallest value
Interquartile range	$IQR = Q_3 - Q_1$	$Q_1, Q_3$ = 25th and 75th percentiles (first and third quartiles)
Lower / upper outlier fences	$Q_1 - 1.5\cdot IQR$ and $Q_3 + 1.5\cdot IQR$	a value beyond either fence is a flagged outlier

Five-number summary: minimum, $Q_1$ , median, $Q_3$ , maximum — the five values that a boxplot draws. The median is the middle value of the sorted data; $Q_1$ and $Q_3$ are the medians of the lower and upper halves.

2Probability rules (Ch 3–4)¶

Every probability is a number $0 \le P(A) \le 1$ , where $P(A)$ is the probability of event $A$ .

Rule	Formula	When it applies
Complement rule	$P(A^c) = 1 - P(A)$	$A^c$ = “ $A$ does not happen”; always valid
Addition rule (general)	$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$	any two events; the last term removes double-counting
Addition rule (disjoint)	$P(A \text{ or } B) = P(A) + P(B)$	only when $A$ and $B$ are disjoint (cannot both occur)
Multiplication rule (independent)	$P(A \text{ and } B) = P(A) \times P(B)$	only when $A$ and $B$ are independent

3Discrete random variables (Ch 4–5)¶

A random variable $X$ assigns a number to each outcome; $P(x)$ is the probability $X$ takes the value $x$ .

Quantity	Formula	Symbols
Valid distribution	$\sum P(x) = 1$ , with each $0 \le P(x) \le 1$	the probabilities of all values sum to exactly 1
Expected value (mean)	$E(X) = \mu = \sum x \cdot P(x)$	the probability-weighted (long-run) average of $X$
Standard deviation	$SD(X) = \sigma = \sqrt{\sum (x - \mu)^2 \, P(x)}$	the typical distance of an outcome from $\mu$

4The Normal model (Ch 4)¶

Quantity	Formula	Symbols
z-score	$z = \dfrac{x - \mu}{\sigma}$	how many standard deviations $x$ is above ( $+$ ) or below ( $-$ ) the mean
Empirical rule (68–95–99.7)	68% within $\pm 1\sigma$ , 95% within $\pm 2\sigma$ , 99.7% within $\pm 3\sigma$	of the mean, for an approximately Normal distribution

$\mu$ = population (or model) mean and $\sigma$ = population standard deviation. A Normal model is written $N(\mu, \sigma)$ ; the standard Normal is $N(0,1)$ . To standardize a sample value instead, use $z = (x - \bar{x})/s$ .

5Foundations for inference (Ch 5)¶

Quantity	Formula	Symbols
Standard error of a proportion	$SE_{\hat{p}} = \sqrt{\dfrac{\hat{p}\,(1 - \hat{p})}{n}}$	$\hat{p}$ = sample proportion; typical wobble of $\hat{p}$
Standard error of a mean	$SE_{\bar{x}} = \dfrac{s}{\sqrt{n}}$	use $\sigma/\sqrt{n}$ when $\sigma$ is known
Confidence interval (general)	$\text{point estimate} \pm (\text{critical value}) \times SE$	a range of plausible values for the parameter
Margin of error	$ME = (\text{critical value}) \times SE$	half the width of a confidence interval
Test statistic (general)	$\dfrac{\text{estimate} - \text{null value}}{SE}$	how far the estimate sits from the claim, in $SE$ units
Decision rule	reject $H_0$ if p-value $< \alpha$ ; otherwise fail to reject	$\alpha$ = significance level, chosen before seeing data

6Inference for a proportion (Ch 6)¶

Quantity	Formula	Symbols
Success–failure condition	interval: $n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$ ; test: $np_0 \ge 10$ and $n(1-p_0) \ge 10$	must hold to use a Normal-based procedure
One-proportion z-interval	$\hat{p} \pm z^{*}\sqrt{\dfrac{\hat{p}\,(1-\hat{p})}{n}}$	$SE$ uses the sample value $\hat{p}$
One-proportion z-test	$z = \dfrac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}$	$SE$ uses the claimed value $p_0$

$p$ = the true population proportion (a parameter); $p_0$ = the specific value of $p$ claimed by $H_0$ .

7Inference for a mean (Ch 7)¶

Quantity	Formula	Symbols
One-mean t-interval	$\bar{x} \pm t^{*}\dfrac{s}{\sqrt{n}}$ , with $df = n - 1$	uses the t-distribution because $\sigma$ is unknown
One-mean t-test	$t = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}$ , with $df = n - 1$	$\mu_0$ = the mean claimed by $H_0$

$\mu$ = the true population mean (a parameter); $df$ = degrees of freedom, which fix the exact shape of the t-distribution.

8Critical values¶

z-critical values $z^{*}$ (for Normal-based intervals and tests — short enough to memorize):

Confidence level	90%	95%	99%
$z^{*}$	1.645	1.960	2.576

t-critical values $t^{*}$ are read from a t-table (or invT) using the row for your degrees of freedom, $df = n - 1$ . They are always larger than the matching $z^{*}$ for small samples and shrink toward $z^{*}$ as $df$ grows. For example, the row for $df = 23$ reads:

$df$	90%	95%	99%
23	1.714	2.069	2.807

9Symbols at a glance¶

Symbol	Meaning
$n$	sample size
$\sum$	“add up all the terms”
$\bar{x}$	sample mean (a statistic)
$\mu$	population mean / expected value (a parameter)
$s$ , $s^2$	sample standard deviation and sample variance
$\sigma$ , $\sigma^2$	population standard deviation and variance
$Q_1, Q_3$	first and third quartiles (25th, 75th percentiles)
$IQR$	interquartile range, $Q_3 - Q_1$
$P(A)$ , $A^c$	probability of event $A$ ; complement of $A$
$X$ , $P(x)$	a random variable; the probability it equals the value $x$
$E(X)$ , $SD(X)$	expected value and standard deviation of $X$
$z$	z-score, $(x-\mu)/\sigma$
$p$ , $\hat{p}$	population proportion (parameter); sample proportion (statistic)
$SE$	standard error (the SD of a sampling distribution)
$z^{}$ , $t^{}$	critical values for a chosen confidence level
$ME$	margin of error
$df$	degrees of freedom, $n - 1$ for one-mean inference
$H_0$ , $H_A$	null and alternative hypotheses
$\alpha$	significance level (chosen before seeing the data)
$p_0$ , $\mu_0$	the value claimed under $H_0$ for a proportion or a mean