api: YAML JSON TOON Database
version: 1.0.0
format: json
dataset:
  slug: statistics-basics
  title: Statistics Basics
  description: Fundamental statistical measures and concepts.
  summary: "The basic measures of central tendency, spread and association: mean, median, mode, variance, standard deviation, correlation and the z-score, plus range, interquartile range, covariance, skewness, standard error, coefficient of variation, confidence intervals and the p-value, each with its formula."
  use_when:
    - You need the definition and formula for a descriptive statistic.
    - You are choosing a summary that is robust to outliers.
  caveats:
    - Sample and population versions differ (n versus n - 1 in the denominator).
    - A p-value is not the probability that a hypothesis is true, and correlation does not establish causation.
  related:
    - combinatorics-fundamentals
    - ml-evaluation-metrics
    - set-theory-basics
  category: Mathematics
  category_slug: mathematics
  tags: math,statistics,probability,data-science
  view_count: 0
  entry_count: 15
  fields:
    - section: data
      field: measure
      type: string
      example: Mean (Average)
    - section: data
      field: description
      type: string
      example: Sum of all values divided by count.
    - section: data
      field: formula
      type: string
      example: xbar = Sum(xi) / n
data:
  - measure: Mean (Average)
    description: Sum of all values divided by count.
    formula: xbar = Sum(xi) / n
  - measure: Median
    description: Middle value when data is sorted. Robust to outliers.
    formula: Middle element (or avg of two)
  - measure: Mode
    description: Most frequently occurring value in the dataset.
    formula: argmax frequency
  - measure: Variance
    description: Average squared deviation from the mean.
    formula: sigma2 = Sum(xi - xbar)2 / n
  - measure: Standard Deviation
    description: Square root of variance; measures spread in original units.
    formula: sigma = sqrt(variance)
  - measure: Correlation
    description: Measure of linear relationship between two variables.
    formula: r = Cov(X,Y) / (sigma_x * sigma_y)
  - measure: Z-Score
    description: Number of standard deviations a value is from the mean.
    formula: z = (x - xbar) / sigma
  - measure: Range
    description: Difference between the largest and smallest value; very sensitive to outliers.
    formula: max - min
  - measure: Interquartile Range
    description: Spread of the middle half of the data; robust to outliers.
    formula: Q3 - Q1
  - measure: Covariance
    description: How two variables vary together; positive when they tend to move in the same direction.
    formula: cov(X, Y) = E[(X - mean(X)) (Y - mean(Y))]
  - measure: Skewness
    description: Asymmetry of a distribution; positive when the right tail is longer.
    formula: E[((X - mean) / sd)^3]
  - measure: Standard Error of the Mean
    description: Standard deviation of the sample mean across repeated samples.
    formula: sd / sqrt(n)
  - measure: Coefficient of Variation
    description: Standard deviation relative to the mean, allowing comparison across different scales.
    formula: sd / mean
  - measure: Confidence Interval for a Mean
    description: A range computed so that, over repeated samples, it contains the true mean a stated share of the time.
    formula: mean +/- z * sd / sqrt(n)
  - measure: p-value
    description: Probability of data at least as extreme as observed if the null hypothesis were true.
    formula: P(data at least this extreme | H0)