Search & glossary

Every rule and concept in Statistics & Data, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (36)

68–95–99.7: For a normal (bell-shaped) distribution, about $68\%$ of values lie within $1$ SD of the mean, $95\%$ within $2$ , and $99.7\%$ within $3$ .; The Normal Distribution › The Empirical Rule and the Table
Areas from $\Phi(z)$: The table gives $\Phi(z) = P(Z < z)$ for the standard normal $Z$ .; The Normal Distribution › Using the Standard Normal Table
Averages become normal: Whatever the shape of the population, the distribution of the sample mean $\bar x$ of $n$ independent observations gets closer to a normal curve as $n$ grows, with mean $\mu$ and SD $\sigma/\sqrt n$ .; Sampling Distributions › The Central Limit Theorem
Binomial setting: Fixed number $n$ of independent trials, each a success with the same probability $p$ .; Binomial and Geometric › The Binomial Distribution
Cause needs an experiment: An observational study records what happens; an experiment assigns treatments.; Sampling and Experiments › Experiments versus Observation
Correlation $r$: A scatter plot shows pairs $(x, y)$ .; Two Variables › Scatter Plots and Correlation
Correlation is not causation: A strong $r$ says the variables move together.; Two Variables › Scatter Plots and Correlation
Estimate $\pm$ margin: A confidence interval is $\text{estimate} \pm z^*\cdot\text{SE}$ .; Confidence Intervals › Intervals for a Proportion
Every value visible: For a small quantitative data set, a dot plot puts one dot per value above a number line, stacking repeats.; Data and Displays › Dot Plots and Stem-and-Leaf Plots
Expected value and variance: A random variable $X$ assigns a number to each outcome; its probability distribution lists each value with its probability (summing to $1$ ).; Random Variables › Discrete Random Variables
Fair, favourable, unfavourable: A game is fair when the expected net gain is $0$ .; Random Variables › Expected Value in Decisions
Fences at $1.5\,\text{IQR}$: The fences are $Q_1 - 1.5\,\text{IQR}$ and $Q_3 + 1.5\,\text{IQR}$ .; Percentiles and Box Plots › Outliers and the Fences
Five numbers: The five-number summary is minimum, $Q_1$ , median, $Q_3$ , maximum.; Percentiles and Box Plots › Quartiles and the Five-Number Summary
How good is the line?: $r^2$ is the fraction of the variation in $y$ explained by the line: $r^2 = 1 - \dfrac{\sum(y - \hat y)^2}{\sum(y - \bar y)^2}$ .; Two Variables › Residuals and $r^2$
Linear changes of units: Adding $c$ to every value shifts the mean and median by $c$ and leaves every measure of spread unchanged.; Center and Spread › Shifting and Scaling
Linear combinations: For any random variables, $E[aX + bY] = aE[X] + bE[Y]$ .; Random Variables › Combining Random Variables
Mean and median: The mean $\bar x = \dfrac{\sum x_i}{n}$ is the fair-share average.; Center and Spread › Measures of Center
Means of samples: Take a random sample of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$ .; Sampling Distributions › The Sampling Distribution of a Mean
Null, alternative, $p$ -value: A test starts from the null hypothesis $H_0$ (nothing is going on: $p = 0.5$ , $\mu = 100$ ) and an alternative $H_a$ .; Hypothesis Tests › The Logic of a Test
Planning $n$: The margin of error for a proportion is $m = z^*\sqrt{p(1 - p)/n}$ .; Confidence Intervals › Margin of Error and Sample Size
Population, sample, bias: A population is everyone you want to describe; a sample is the part you actually measure.; Sampling and Experiments › Samples and Bias
Proportions: A sample proportion $\hat p$ of successes has mean $p$ and standard deviation $\sqrt{\dfrac{p(1-p)}{n}}$ , approximately normal when $np \ge 10$ and $n(1-p) \ge 10$ .; Sampling Distributions › The Sampling Distribution of a Proportion
Range, IQR, standard deviation: The range is max minus min.; Center and Spread › Measures of Spread
Reading a histogram: A histogram groups values into equal-width bins and draws a bar whose height is the count in each bin.; Data and Displays › Histograms and Shape
Regression line: The least-squares line $\hat y = a + bx$ minimizes the sum of squared vertical residuals $y - \hat y$ .; Two Variables › The Least-Squares Line
Skew points to the tail: Skewed right means the long tail is on the right (a few large values).; Data and Displays › Histograms and Shape
Standardizing: A z-score says how many standard deviations a value sits from the mean: $z = \dfrac{x - \mu}{\sigma}$ .; The Normal Distribution › Z-Scores
Stratified, cluster, systematic: A stratified sample divides the population into groups (strata) and takes a random sample within each, so every group is represented.; Sampling and Experiments › Sampling Designs
The $1.5\,\text{IQR}$ rule: A value is an outlier if it lies below $Q_1 - 1.5\,\text{IQR}$ or above $Q_3 + 1.5\,\text{IQR}$ .; Percentiles and Box Plots › Quartiles and the Five-Number Summary
The $p$ th percentile: The $p$ th percentile is the value below which $p\%$ of the data falls.; Percentiles and Box Plots › Percentiles
The $t$ interval: For a mean with unknown $\sigma$ : $\bar x \pm t^*\dfrac{s}{\sqrt n}$ , where $t^*$ comes from the $t$ distribution with $n - 1$ degrees of freedom.; Confidence Intervals › Intervals for a Mean
The $t$ test: For a mean: $t = \dfrac{\bar x - \mu_0}{s/\sqrt n}$ with $n - 1$ degrees of freedom.; Hypothesis Tests › Tests for Means and Two Samples
Two kinds of data: A variable is categorical when it sorts things into groups (eye color, favorite sport) and quantitative when it measures an amount (height, test score).; Data and Displays › Categorical and Quantitative
Two ways to be wrong: A Type I error rejects a true $H_0$ (probability $\alpha$ ).; Hypothesis Tests › Errors and Power
Waiting for the first success: Repeat independent trials with success probability $p$ until the first success.; Binomial and Geometric › The Geometric Distribution
When the bars look like a bell: When $np \ge 10$ and $n(1-p) \ge 10$ , the binomial is close to normal with the same mean $np$ and SD $\sqrt{np(1-p)}$ , so z-scores answer "at least $k$ " questions without summing dozens of terms.; Binomial and Geometric › The Normal Approximation