StatsMasters — Statistics Cheat Sheet
https://statsmasters.com/resources/statistics-cheat-sheet
Version: October 2026

SUMMARIZE A SAMPLE
Mean: x-bar = sum(x_i) / n
Sample variance: s^2 = sum((x_i - x-bar)^2) / (n - 1)
Sample SD: s = sqrt(s^2)
Standard error of mean: SE = s / sqrt(n)
For the complete population, variance divides by n instead of n - 1.

STANDARDIZE A VALUE
z = (x - mu) / sigma
Under a normal model: left area = Phi(z); right area = 1 - Phi(z).
Between a and b: Phi(b) - Phi(a).
Standardized non-normal data need not have normal probabilities.

CONFIDENCE INTERVAL FOR A MEAN
Unknown population SD: x-bar +/- t* s / sqrt(n), df = n - 1.
Known population SD: x-bar +/- z* sigma / sqrt(n).
Check independence and appropriate distribution assumptions.
95% refers to long-run coverage of the procedure over repeated samples.

MEAN TESTS
One-sample t = (x-bar - mu_0) / (s / sqrt(n))
Welch t = (x-bar_1 - x-bar_2) / sqrt(s_1^2/n_1 + s_2^2/n_2)
Welch df is method-specific and can be fractional.
Paired t = mean(differences) / (SD(differences) / sqrt(number of pairs))
A p-value is not the probability the null hypothesis is true.

CHI-SQUARE AND ANOVA
Chi-square = sum((observed - expected)^2 / expected)
Contingency expected count = row total * column total / grand total.
Independence test df = (rows - 1) * (columns - 1).
ANOVA F = MS_between / MS_within; df1 = groups - 1; df2 = N - groups.
Check independent observations, expected counts, and model assumptions.
Standard one-way ANOVA assumes equal variances and suitable errors.

PROBABILITY
Complement: P(not A) = 1 - P(A)
Union: P(A or B) = P(A) + P(B) - P(A and B)
Conditional: P(A given B) = P(A and B) / P(B), with P(B) > 0.
Independence: P(A and B) = P(A) * P(B).
Mutually exclusive is not the same as independent.

CENTRAL NORMAL INTERVALS (positive z cutoff)
90%: 1.645; 95%: 1.960; 99%: 2.576.
95th percentile: 1.645; 97.5th percentile: 1.960.

Start with your question, check assumptions, and report effect size and
uncertainty alongside statistical significance.