Statistics formula sheet
Essential formulas and reminders for your next statistics problem. Print this page or choose “Save as PDF” in your browser’s print dialog.
Summarize a sample
- Mean
- x̄ = Σxᵢ / n
- Sample variance
- s² = Σ(xᵢ − x̄)² / (n − 1)
- Sample standard deviation
- s = √s²
- Standard error of the mean
- SE = s / √n
Use n in the variance denominator for the complete population. Standard error measures uncertainty in the estimated mean, not the spread of individual values.
Use this in practice →Standardize a value
- Z-score
- z = (x − μ) / σ
- Area to the left
- P(Z ≤ z) = Φ(z)
- Area to the right
- P(Z > z) = 1 − Φ(z)
- Area between a and b
- Φ(b) − Φ(a)
The probability statements assume a standard normal model. A standardized value from another distribution need not have normal probabilities.
Use this in practice →Estimate a mean
- t interval, unknown σ
- x̄ ± t* × s / √n
- Degrees of freedom
- df = n − 1
- z interval, known σ
- x̄ ± z* × σ / √n
Check independence and distribution assumptions. A 95% procedure covers the true mean in about 95% of repeated samples; it is not a 95% probability claim about this fixed parameter.
Use this in practice →Test a mean
- One-sample t
- t = (x̄ − μ₀) / (s / √n)
- Welch two-sample t
- t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)
- Paired t
- t = d̄ / (s_d / √n)
For paired data, work with within-pair differences. Welch’s test has method-specific, often fractional degrees of freedom. A p-value is not the probability that the null is true.
Use this in practice →Categorical data & ANOVA
- Chi-square statistic
- χ² = Σ(O − E)² / E
- Independence test df
- (rows − 1) × (columns − 1)
- One-way ANOVA F
- F = MS_between / MS_within
- ANOVA df₁, df₂
- k − 1, N − k
For contingency tables, E = row total × column total / grand total. Check expected counts. Standard one-way ANOVA assumes equal variances and independent, suitable errors.
Use this in practice →Probability rules
- Complement
- P(Aᶜ) = 1 − P(A)
- Union
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- Conditional
- P(A | B) = P(A ∩ B) / P(B)
- Independent events
- P(A ∩ B) = P(A) × P(B)
Conditional probability requires P(B) > 0. Mutually exclusive events cannot happen together; independent events do not change one another’s probabilities.
Use this in practice →Common normal cutoffs
| Central confidence level | Two-tailed α | Positive z cutoff |
|---|---|---|
| 90% | 0.10 | 1.645 |
| 95% | 0.05 | 1.960 |
| 99% | 0.01 | 2.576 |
These are central normal intervals. The 95th percentile alone is z ≈ 1.645; the upper end of a central 95% interval is the 97.5th percentile, z ≈ 1.960.
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