What is the primary difference between descriptive and inferential statistics?

Descriptive statistics summarize sample data; inferential statistics make statements about populations.

What does a z-score represent?

The distance of a data point from the mean in standard deviation units.

Statistical Concepts Cheat Sheet

Statistics is the science of collecting, organizing, summarizing, and analyzing data to draw conclusions with confidence. It involves understanding populations, samples, variables, and employing various methods for data collection, summarization, and inference.

Core Principles

Statistics involves descriptive methods (summarizing data) and inferential methods (making statements about populations from samples).
A population is the entire group of interest, while a sample is a subset drawn from it. Parameters describe populations, statistics describe samples.
Variables can be qualitative (categorical) or quantitative (numeric), and quantitative variables can be discrete (countable) or continuous (interval).
Levels of measurement (Nominal, Ordinal, Interval, Ratio) dictate the types of analysis possible.
Data collection methods include observational studies and designed experiments, each with specific types and sampling techniques.
Bias in sampling occurs when a sample does not represent the target population, stemming from issues like undercoverage or non-response.

Formulas

$ \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i \quad \text{(sample mean)} $
$ \mu = \frac{1}{N}\sum_{i=1}^{N} X_i \quad \text{(population mean)} $
$ R = \max(x_i) - \min(x_i) \quad \text{(range)} $
$ \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(X_i - \mu)^2} \quad \text{(population standard deviation)} $
$ s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2} \quad \text{(sample standard deviation)} $
$ z = \frac{x - \mu}{\sigma} \quad \text{(z-score)} $
$ r = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x})^2 \, \sum (y_i-\bar{y})^2}} \quad \text{(correlation coefficient)} $
$ \hat{y} = b_1 x + b_0 \quad \text{(predicted response)} $
$ b_1 = r \frac{s_y}{s_x} \quad \text{(slope of regression line)} $
$ b_0 = \bar{y} - b_1 \bar{x} \quad \text{(intercept of regression line)} $
$ P(E \cup F) = P(E) + P(F) \quad \text{(disjoint events)} $
$ P(E \cup F) = P(E) + P(F) - P(E \cap F) \quad \text{(general addition rule)} $
$ P(E^c) = 1 - P(E) \quad \text{(complement rule)} $
$ P(E \cap F) = P(E) \, P(F) \quad \text{(independent events)} $
$ P(F \mid E) = \frac{P(E \cap F)}{P(E)} \quad \text{(conditional probability)} $
$ ^nP_r = \frac{n!} {(n-r)!} \quad \text{(permutations)} $
$ ^nC_r = \frac{n!} {r!(n-r)!} \quad \text{(combinations)} $
$ \mu = E(X) = \sum x \, P(X = x) \quad \text{(expected value)} $
$ \sigma = \sqrt{\sum (x - \mu)^2 \, P(X = x)} \quad \text{(standard deviation of RV)} $
$ P(X = x) = \binom{n}{x} p^{x} (1-p)^{\,n-x} \quad \text{(binomial PMF)} $
$ \mu = np \quad \text{(binomial mean)} $
$ \sigma = \sqrt{np(1-p)} \quad \text{(binomial standard deviation)} $
$ P(X = x) = \frac{e^{-\lambda}\lambda^{x}}{x!} \quad \text{(Poisson PMF)} $
$ Z = \frac{x - \mu}{\sigma} \quad \text{(standardization)} $
$ \mu_{\bar{X}} = \mu \quad \text{(mean of sample mean)} $
$ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \quad \text{(standard error of sample mean)} $
$ \mu_{\hat{p}} = p \quad \text{(mean of sample proportion)} $
$ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \quad \text{(standard error of sample proportion)} $
$ E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \quad \text{(margin of error for proportion CI)} $
$ n = \left(\frac{z_{\alpha/2}}{E}\right)^{\!2}\hat{p}(1-\hat{p}) \quad \text{(sample size for proportion)} $
$ E = t_{\alpha/2, n-1}\frac{s}{\sqrt{n}} \quad \text{(margin of error for mean CI, unknown } \sigma \text{)} $
$ n = \left(\frac{z_{\alpha/2}\,s}{E}\right)^{\!2} \quad \text{(sample size for mean)} $
$ \chi^{2} = \frac{(n-1)s^{2}}{\sigma_{0}^{2}} \quad \text{(chi-square test for std dev)} $

Quiz

What is the primary difference between descriptive and inferential statistics?: Descriptive statistics summarize sample data; inferential statistics make statements about populations.
Which level of measurement has a true zero, allowing for meaningful ratios?: Ratio
What does a z-score represent?: The distance of a data point from the mean in standard deviation units.