Statistics Formulas
A copy-able sheet — descriptive measures, probability, distributions, intervals, tests, regression, and effect sizes.
Showing 48 of 48. Click any formula to copy it.
Sample statistics use Latin letters (x̄, s, p̂) and population parameters use Greek ones (μ, σ, p). Where a formula divides by n − 1 it is estimating a population value from a sample; dividing by n treats the data as the whole population.
Most of an introductory statistics course runs on a few dozen formulas. The sample mean x̄ = Σxᵢ ÷ n and variance s² = Σ(xᵢ − x̄)² ÷ (n − 1) underpin nearly everything after them, and every confidence interval has the same shape: estimate ± critical value × standard error.
One pattern behind most of the sheet
Confidence intervals and hypothesis tests look like a long list of separate formulas, but they are two views of one idea. An interval is estimate ± critical value × standard error; a test statistic is (estimate − hypothesised value) ÷ standard error. Once you can identify the estimate and its standard error, the rest of the formula writes itself, whether the quantity is a mean, a proportion, or a difference between two groups.
The standard error is the piece that changes. For a mean it is σ ÷ √n, for a proportion √(p(1 − p) ÷ n), and for a difference of means it combines both samples through the pooled variance. Every one of them shrinks as √n grows, which is the formal statement of why larger samples give tighter answers — and why quadrupling the sample only halves the error.
Latin letters and Greek ones
The notation carries meaning. Latin letters describe a sample you measured: x̄, s, p̂. Greek letters describe the population you are inferring about: μ, σ, p. The whole of inferential statistics is the business of using the first to make defensible claims about the second, so mixing them up in a formula usually means the wrong quantity is being estimated.
Sample against population notation
Which symbol a formula uses tells you whether it describes data in hand or the population behind it.
| Quantity | Sample (statistic) | Population (parameter) |
|---|---|---|
| Mean | x̄ | μ |
| Standard deviation | s | σ |
| Variance | s² | σ² |
| Proportion | p̂ | p |
| Size | n | N |
| Correlation | r | ρ |
Which test goes with which question
The degrees of freedom column is what you carry into the matching table of critical values.
| Question | Test | Degrees of freedom |
|---|---|---|
| Does one mean differ from a known value? | One-sample t-test | n − 1 |
| Do two independent group means differ? | Pooled two-sample t-test | n₁ + n₂ − 2 |
| Did the same subjects change? | Paired t-test | pairs − 1 |
| Do three or more group means differ? | One-way ANOVA | groups − 1 and n − groups |
| Do observed counts match expected ones? | Chi-square goodness of fit | categories − 1 |
| Are two categorical variables related? | Chi-square independence | (rows − 1) × (columns − 1) |
Where the formulas are misapplied
The commonest error is dividing by n instead of n − 1 when computing a sample standard deviation. The n − 1 correction exists because the sample mean is itself estimated from the same data, which makes the raw spread slightly too small; dividing by n understates the variability of every quantity built on it.
Two more are worth watching. R² = r² holds only in simple linear regression with one predictor — with several, R² has to come from the sums of squares. And a chi-square test works on raw counts alone: converting the table to percentages first changes the effective sample size and invalidates the result.