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Statistics Formulas

Core statistics formulas and what each one calculates.

84 cardsGrades 9–12, College#statistics#formulas#math

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  • Sample meanName
    x-bar = (sum of x) / n
  • Population meanName
    mu = (sum of x) / N
  • Median of an odd countName
    The middle value when sorted
  • Median of an even countName
    The average of the two middle values
  • RangeName
    Maximum minus minimum
  • Interquartile rangeName
    IQR = Q3 - Q1
  • Upper outlier fenceName
    Q3 + 1.5 x IQR
  • Lower outlier fenceName
    Q1 - 1.5 x IQR
  • Sample varianceName
    s^2 = sum of (x - x-bar)^2 / (n - 1)
  • Population varianceName
    sigma^2 = sum of (x - mu)^2 / N
  • Sample standard deviationName
    s = square root of sample variance
  • Z-scoreName
    z = (x - mean) / standard deviation
  • Standard error of the meanName
    s / sqrt(n)
  • Standard error of a proportionName
    sqrt(p(1 - p) / n)
  • Coefficient of variationName
    (standard deviation / mean) x 100 percent
  • Percentile rankName
    (number of values below x / total values) x 100
  • Probability of an eventName
    Favorable outcomes / total outcomes
  • Complement ruleName
    P(A') = 1 - P(A)
  • Addition ruleName
    P(A or B) = P(A) + P(B) - P(A and B)
  • Multiplication rule, independentName
    P(A and B) = P(A) x P(B)
  • Conditional probabilityName
    P(A|B) = P(A and B) / P(B)
  • Bayes' theoremName
    P(A|B) = P(B|A) x P(A) / P(B)
  • PermutationsName
    nPr = n! / (n - r)!
  • CombinationsName
    nCr = n! / (r!(n - r)!)
  • FactorialName
    n! = n x (n-1) x ... x 1
  • Expected valueName
    E(X) = sum of x x P(x)
  • Variance of a random variableName
    sum of (x - mu)^2 x P(x)
  • Binomial probabilityName
    P(X = k) = nCk x p^k x (1 - p)^(n - k)
  • Binomial meanName
    mu = np
  • Binomial standard deviationName
    sigma = sqrt(np(1 - p))
  • Geometric probabilityName
    P(first success on trial k) = (1 - p)^(k - 1) x p
  • Poisson probabilityName
    P(X = k) = e^(-lambda) x lambda^k / k!
  • Normal approx. to binomial conditionName
    np >= 10 and n(1 - p) >= 10
  • Empirical ruleName
    68, 95, 99.7 percent within 1, 2, 3 standard deviations
  • Chebyshev's inequalityName
    At least 1 - 1/k^2 of data lie within k standard deviations
  • Confidence interval for a meanName
    x-bar +/- t* x s/sqrt(n)
  • Confidence interval for a proportionName
    p-hat +/- z* x sqrt(p-hat(1 - p-hat)/n)
  • Margin of error (proportion)Name
    z* x sqrt(p-hat(1 - p-hat)/n)
  • Sample size for a proportionName
    n = (z*/E)^2 x p(1 - p)
  • One-sample t statisticName
    t = (x-bar - mu0) / (s / sqrt(n))
  • One-sample z statistic for proportionName
    z = (p-hat - p0) / sqrt(p0(1 - p0)/n)
  • Two-sample t statisticName
    t = (x1-bar - x2-bar) / sqrt(s1^2/n1 + s2^2/n2)
  • Degrees of freedom, one sampleName
    n - 1
  • Chi-square statisticName
    sum of (observed - expected)^2 / expected
  • Expected count in a tableName
    (row total x column total) / grand total
  • Degrees of freedom, chi-square tableName
    (rows - 1)(columns - 1)
  • Pearson correlationName
    r = (1/(n-1)) x sum of (z_x x z_y)
  • Regression lineName
    y-hat = a + bx
  • Slope of the regression lineName
    b = r x (s_y / s_x)
  • Intercept of the regression lineName
    a = y-bar - b x x-bar
  • ResidualName
    observed y - predicted y
  • Coefficient of determinationName
    r^2
  • Sum of squares totalName
    sum of (y - y-bar)^2
  • ANOVA F statisticName
    F = mean square between / mean square within
  • Effect size, Cohen's dName
    (mean1 - mean2) / pooled standard deviation
  • Type I error probabilityName
    Alpha, the significance level
  • PowerName
    1 - beta
  • OddsName
    p / (1 - p)
  • Odds ratioName
    (a/b) / (c/d) for a 2 by 2 table
  • Relative riskName
    Risk in exposed group / risk in unexposed group
  • Percent changeName
    (new - old) / old x 100
  • Weighted meanName
    sum of (w x x) / sum of w
  • Geometric meanName
    nth root of the product of n values
  • Harmonic meanName
    n / sum of (1/x)
  • ModeName
    The most frequent value
  • Skewness directionName
    Mean greater than median suggests right skew
  • Mean absolute deviationName
    sum of |x - mean| / n
  • Law of total probabilityName
    P(B) = sum of P(B|A_i) x P(A_i)
  • Hypergeometric probabilityName
    P(X = k) = [C(K,k) x C(N-K, n-k)] / C(N,n)
  • Central limit theorem standard deviationName
    sigma / sqrt(n)
  • Finite population correctionName
    sqrt((N - n)/(N - 1))
  • Pooled proportionName
    (x1 + x2) / (n1 + n2)
  • Z critical value 95%Name
    1.96
  • Z critical value 99%Name
    2.576
  • Z critical value 90%Name
    1.645
  • Cumulative relative frequencyName
    Running total of relative frequencies
  • Class widthName
    (max - min) / number of classes
  • Sturges' ruleName
    Number of classes is about 1 + log2(n)
  • Standard score for sample meanName
    z = (x-bar - mu) / (sigma / sqrt(n))
  • Tukey fencesName
    Boundaries for outliers based on the IQR
  • Lower quartile positionName
    (n + 1) / 4
  • Rule for sample variance denominatorName
    Use n - 1 for samples to correct bias
  • Joint probability, dependentName
    P(A and B) = P(A) x P(B|A)
  • Mutually exclusive eventsName
    P(A and B) = 0

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