{
 "source": "https://peabody.kids/flashcards/tp-stats-formulas",
 "id": "tp-stats-formulas",
 "title": "Statistics Formulas",
 "category": "test-prep",
 "subject": "Statistics",
 "levels": [
  "9-12",
  "college"
 ],
 "description": "Core statistics formulas and what each one calculates.",
 "tags": [
  "statistics",
  "formulas",
  "math"
 ],
 "frontLabel": "Name",
 "backLabel": "Formula",
 "cards": [
  [
   "Sample mean",
   "x-bar = (sum of x) / n"
  ],
  [
   "Population mean",
   "mu = (sum of x) / N"
  ],
  [
   "Median of an odd count",
   "The middle value when sorted"
  ],
  [
   "Median of an even count",
   "The average of the two middle values"
  ],
  [
   "Range",
   "Maximum minus minimum"
  ],
  [
   "Interquartile range",
   "IQR = Q3 - Q1"
  ],
  [
   "Upper outlier fence",
   "Q3 + 1.5 x IQR"
  ],
  [
   "Lower outlier fence",
   "Q1 - 1.5 x IQR"
  ],
  [
   "Sample variance",
   "s^2 = sum of (x - x-bar)^2 / (n - 1)"
  ],
  [
   "Population variance",
   "sigma^2 = sum of (x - mu)^2 / N"
  ],
  [
   "Sample standard deviation",
   "s = square root of sample variance"
  ],
  [
   "Z-score",
   "z = (x - mean) / standard deviation"
  ],
  [
   "Standard error of the mean",
   "s / sqrt(n)"
  ],
  [
   "Standard error of a proportion",
   "sqrt(p(1 - p) / n)"
  ],
  [
   "Coefficient of variation",
   "(standard deviation / mean) x 100 percent"
  ],
  [
   "Percentile rank",
   "(number of values below x / total values) x 100"
  ],
  [
   "Probability of an event",
   "Favorable outcomes / total outcomes"
  ],
  [
   "Complement rule",
   "P(A') = 1 - P(A)"
  ],
  [
   "Addition rule",
   "P(A or B) = P(A) + P(B) - P(A and B)"
  ],
  [
   "Multiplication rule, independent",
   "P(A and B) = P(A) x P(B)"
  ],
  [
   "Conditional probability",
   "P(A|B) = P(A and B) / P(B)"
  ],
  [
   "Bayes' theorem",
   "P(A|B) = P(B|A) x P(A) / P(B)"
  ],
  [
   "Permutations",
   "nPr = n! / (n - r)!"
  ],
  [
   "Combinations",
   "nCr = n! / (r!(n - r)!)"
  ],
  [
   "Factorial",
   "n! = n x (n-1) x ... x 1"
  ],
  [
   "Expected value",
   "E(X) = sum of x x P(x)"
  ],
  [
   "Variance of a random variable",
   "sum of (x - mu)^2 x P(x)"
  ],
  [
   "Binomial probability",
   "P(X = k) = nCk x p^k x (1 - p)^(n - k)"
  ],
  [
   "Binomial mean",
   "mu = np"
  ],
  [
   "Binomial standard deviation",
   "sigma = sqrt(np(1 - p))"
  ],
  [
   "Geometric probability",
   "P(first success on trial k) = (1 - p)^(k - 1) x p"
  ],
  [
   "Poisson probability",
   "P(X = k) = e^(-lambda) x lambda^k / k!"
  ],
  [
   "Normal approx. to binomial condition",
   "np >= 10 and n(1 - p) >= 10"
  ],
  [
   "Empirical rule",
   "68, 95, 99.7 percent within 1, 2, 3 standard deviations"
  ],
  [
   "Chebyshev's inequality",
   "At least 1 - 1/k^2 of data lie within k standard deviations"
  ],
  [
   "Confidence interval for a mean",
   "x-bar +/- t* x s/sqrt(n)"
  ],
  [
   "Confidence interval for a proportion",
   "p-hat +/- z* x sqrt(p-hat(1 - p-hat)/n)"
  ],
  [
   "Margin of error (proportion)",
   "z* x sqrt(p-hat(1 - p-hat)/n)"
  ],
  [
   "Sample size for a proportion",
   "n = (z*/E)^2 x p(1 - p)"
  ],
  [
   "One-sample t statistic",
   "t = (x-bar - mu0) / (s / sqrt(n))"
  ],
  [
   "One-sample z statistic for proportion",
   "z = (p-hat - p0) / sqrt(p0(1 - p0)/n)"
  ],
  [
   "Two-sample t statistic",
   "t = (x1-bar - x2-bar) / sqrt(s1^2/n1 + s2^2/n2)"
  ],
  [
   "Degrees of freedom, one sample",
   "n - 1"
  ],
  [
   "Chi-square statistic",
   "sum of (observed - expected)^2 / expected"
  ],
  [
   "Expected count in a table",
   "(row total x column total) / grand total"
  ],
  [
   "Degrees of freedom, chi-square table",
   "(rows - 1)(columns - 1)"
  ],
  [
   "Pearson correlation",
   "r = (1/(n-1)) x sum of (z_x x z_y)"
  ],
  [
   "Regression line",
   "y-hat = a + bx"
  ],
  [
   "Slope of the regression line",
   "b = r x (s_y / s_x)"
  ],
  [
   "Intercept of the regression line",
   "a = y-bar - b x x-bar"
  ],
  [
   "Residual",
   "observed y - predicted y"
  ],
  [
   "Coefficient of determination",
   "r^2"
  ],
  [
   "Sum of squares total",
   "sum of (y - y-bar)^2"
  ],
  [
   "ANOVA F statistic",
   "F = mean square between / mean square within"
  ],
  [
   "Effect size, Cohen's d",
   "(mean1 - mean2) / pooled standard deviation"
  ],
  [
   "Type I error probability",
   "Alpha, the significance level"
  ],
  [
   "Power",
   "1 - beta"
  ],
  [
   "Odds",
   "p / (1 - p)"
  ],
  [
   "Odds ratio",
   "(a/b) / (c/d) for a 2 by 2 table"
  ],
  [
   "Relative risk",
   "Risk in exposed group / risk in unexposed group"
  ],
  [
   "Percent change",
   "(new - old) / old x 100"
  ],
  [
   "Weighted mean",
   "sum of (w x x) / sum of w"
  ],
  [
   "Geometric mean",
   "nth root of the product of n values"
  ],
  [
   "Harmonic mean",
   "n / sum of (1/x)"
  ],
  [
   "Mode",
   "The most frequent value"
  ],
  [
   "Skewness direction",
   "Mean greater than median suggests right skew"
  ],
  [
   "Mean absolute deviation",
   "sum of |x - mean| / n"
  ],
  [
   "Law of total probability",
   "P(B) = sum of P(B|A_i) x P(A_i)"
  ],
  [
   "Hypergeometric probability",
   "P(X = k) = [C(K,k) x C(N-K, n-k)] / C(N,n)"
  ],
  [
   "Central limit theorem standard deviation",
   "sigma / sqrt(n)"
  ],
  [
   "Finite population correction",
   "sqrt((N - n)/(N - 1))"
  ],
  [
   "Pooled proportion",
   "(x1 + x2) / (n1 + n2)"
  ],
  [
   "Z critical value 95%",
   "1.96"
  ],
  [
   "Z critical value 99%",
   "2.576"
  ],
  [
   "Z critical value 90%",
   "1.645"
  ],
  [
   "Cumulative relative frequency",
   "Running total of relative frequencies"
  ],
  [
   "Class width",
   "(max - min) / number of classes"
  ],
  [
   "Sturges' rule",
   "Number of classes is about 1 + log2(n)"
  ],
  [
   "Standard score for sample mean",
   "z = (x-bar - mu) / (sigma / sqrt(n))"
  ],
  [
   "Tukey fences",
   "Boundaries for outliers based on the IQR"
  ],
  [
   "Lower quartile position",
   "(n + 1) / 4"
  ],
  [
   "Rule for sample variance denominator",
   "Use n - 1 for samples to correct bias"
  ],
  [
   "Joint probability, dependent",
   "P(A and B) = P(A) x P(B|A)"
  ],
  [
   "Mutually exclusive events",
   "P(A and B) = 0"
  ]
 ]
}