Earth Stats Week 2-3 Powerpoints

Vocabulary flashcards covering key statistical concepts from lecture notes on Binomial and Normal Distributions, Confidence Intervals, and Parametric Tests.

Binomial Distribution

A probability distribution applying to discrete events for which only two outcomes are possible.

Binomial Formula

The formula used to calculate the probability $P$ of $r$ successes in $n$ events with success probability $p$: $P = \frac{n!}{(n-r)!\,r!} (1-p)^{n-r} p^r$.

Galton Board

A physical device used to demonstrate the binomial distribution, where falling balls make binary left or right decisions at each row of pegs.

Normal (Gaussian) Distribution

A continuous probability distribution defined by a mean ($\mu$) and standard deviation ($\sigma$), symmetric around its mean, denser in the center, with total area under the PDF curve equal to $1.0$.

Mode

The most frequent or probable observation in a dataset; it is applicable to both quantitative and qualitative data and is not influenced by extreme observations.

Median

The central value of a quantitative dataset where $50\%$ of observations lie above and $50\%$ lie below; it is not influenced by extreme observations.

Mean

The arithmetic average of a quantitative dataset calculated as $\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$, which is influenced by extreme observations.

Sample Variance

A computed measure of dispersion around the sample mean defined for a sample as $s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n-1}$.

Standard Normal Distribution (SND)

A specific normal distribution that has a mean of $0$ and a standard deviation of $1$.

Z-score

The value of an observation expressed as the number of standard deviations it lies relative to the mean, calculated as $z_i = \frac{x_i - \bar{x}}{s}$.

Point Estimate

A single numerical sample statistic (such as sample mean $\bar{x}$ or sample proportion $p$) used to estimate an unknown population parameter.

Confidence Interval

An interval estimate providing a range of values based on sample observations and standard error, giving information about closeness to an unknown population parameter at a specified level of confidence.

Standard Error (SE)

A measure of the dispersion of sample means, defined as the standard deviation of sample means, calculated as $SE = \frac{\sigma}{\sqrt{n}}$ (or $\frac{s}{\sqrt{n}}$).

Student's t Distribution

A family of bell-shaped, symmetric probability distributions with fatter tails than the standard normal distribution, used when the population standard deviation $\sigma$ is unknown.

Degrees of Freedom ($d.f.$)

The number of observations that are free to vary after a sample statistic has been calculated, given by $d.f. = n - 1$ for a sample of size $n$.

Parametric Statistical Test

A statistical test assuming data comes from a population described by a specific probability distribution (such as a normal distribution) and evaluating parameters like means and variances.

Nonparametric Statistical Test

A statistical test that makes fewer assumptions about the mathematical form of the population distribution, usually does not require normality, and often evaluates ranks or location differences.

Statistical Power

The probability that a statistical test will correctly reject the null hypothesis ($H_0$) when $H_0$ is false, expressed as $1 - \beta$.

Null Hypothesis ($H_0$)

A statement about a population parameter asserting no significant effect, difference, or relationship, serving as the default starting assumption in hypothesis testing.

Research Hypothesis ($H_r$)

The alternative hypothesis that challenges the status quo and represents the condition or effect the researcher wishes to establish, never containing an equality sign.

Type I Error

The error committed when a true null hypothesis ($H_0$) is rejected; its probability is denoted by $\alpha$ (the level of significance).

Type II Error

The error committed when a false null hypothesis ($H_0$) is failed to be rejected; its probability is denoted by $\beta$.

p-value

The probability of obtaining a test statistic equal to or more extreme than the observed sample value given that the null hypothesis ($H_0$) is true.

F-Test

A statistical test comparing two population variances by evaluating the ratio of sample variances $F = \frac{s_1^2}{s_2^2}$ with degrees of freedom $df_1 = n_1 - 1$ and $df_2 = n_2 - 1$.

Shapiro-Wilk Test

A statistical test used to evaluate whether sample data follows a normal distribution for a given significance level $\alpha$.

Welch's t-test

A variation of the two-sample t-test utilized when the assumption of equal population variances between two groups is violated.