What is the difference between population and sample formulas?

Population formulas (using μ, σ, N) describe the entire group you are studying. Sample formulas (using x̄, s, n) estimate population parameters from a subset. The key difference: sample variance divides by n−1 instead of n — the Bessel correction — to produce an unbiased estimate. Use population formulas when you have data for everyone; use sample formulas when your data is a subset drawn from a larger group.

Which formula should I use for comparing two groups?

To compare means of two groups: use the independent two-sample t-test (Welch version if variances may differ). To compare proportions: use a two-proportion z-test. To compare more than two group means: use ANOVA (F-test). To compare categorical distributions: use chi-square test. The choice depends on (1) whether data is continuous or categorical, (2) how many groups, and (3) whether you know the population standard deviation.

What does standard error measure, and how is it different from standard deviation?

Standard deviation (SD) measures the spread of individual values around the mean — it describes variability in the data. Standard error (SE = SD / √n) measures the precision of the sample mean as an estimate of the population mean — it describes how much the mean varies from sample to sample. SE shrinks as sample size increases; SD does not. Use SD to describe data spread; use SE to describe estimation precision.

What does Pearson's r tell me?

Pearson r measures the strength and direction of the linear relationship between two variables. r = +1: perfect positive linear relationship. r = 0: no linear relationship (though a non-linear relationship may exist). r = −1: perfect negative linear relationship. Rules of thumb: |r| 0.7 strong. Important: r ≠ causation. Also, r only captures linear relationships — a strong curved relationship can have r ≈ 0.

What is the difference between z-test and t-test?

Both test whether a mean differs from a target value. The z-test requires knowing the population standard deviation σ (rare in practice) or having a very large sample (n ≥ 30 where t ≈ z). The t-test uses the estimated sample standard deviation s — appropriate for small samples where this uncertainty matters. The t-distribution has heavier tails than normal to account for the extra uncertainty of estimating σ. For n > 120, the difference is negligible.

Statistics Formulas — Complete Reference Guide

Name: Statistics Formulas — Complete Reference Guide
Availability: OnlineOnly
Author: CalcMulti Editorial Team

By CalcMulti Editorial Team·Updated: February 2026·10 min read

This reference guide collects the core formulas used in descriptive statistics, probability, and inferential statistics. Each formula is presented with its notation, what each symbol means, and a worked example. Bookmark this page as your go-to statistics formula sheet.

Formulas are organised by topic: start with descriptive statistics for summarising data, move to probability for modelling uncertainty, and finish with inferential statistics for drawing conclusions from samples.

Central Tendency

Central tendency measures describe where the centre of a dataset falls. The three main measures are the arithmetic mean, median, and mode — each appropriate for different data types and distributions.

Measure	Formula	Use when	Sensitive to outliers?
Arithmetic Mean	x̄ = Σx / n	Symmetric data, no extreme outliers	Yes — strongly
Weighted Mean	x̄w = Σ(wᵢxᵢ) / Σwᵢ	Values have different importance/frequency	Yes
Geometric Mean	GM = (x₁×x₂×…×xₙ)^(1/n)	Multiplicative data: growth rates, ratios	Less than arithmetic mean
Harmonic Mean	HM = n / Σ(1/xᵢ)	Rates and speeds (distance/time)	Yes — to very small values
Median	Middle value when sorted	Skewed data, ordinal data, outliers present	No — robust
Mode	Most frequent value(s)	Categorical data, multimodal distributions	No

Measures of Spread (Variability)

Spread measures quantify how dispersed values are around the centre. A low spread means values cluster tightly; a high spread means they are widely scattered. The choice of spread measure depends on data type and whether outliers are present.

Measure	Formula	Units	Notes
Range	max − min	Same as data	Simple but heavily influenced by extremes
IQR	Q3 − Q1	Same as data	Middle 50%; robust to outliers
Population Variance	σ² = Σ(xᵢ − μ)² / n	Squared units	Use when data IS the full population
Sample Variance	s² = Σ(xᵢ − x̄)² / (n−1)	Squared units	Use when data is a sample; Bessel's correction
Population SD	σ = √σ²	Same as data	Interpret directly in original units
Sample SD	s = √s²	Same as data	Most common reported measure of spread
Coefficient of Variation	CV = (s / x̄) × 100%	%	Relative spread — compare datasets with different units
Standard Error	SE = s / √n	Same as data	Precision of the sample mean estimate

Position and Standardisation

Position measures locate a specific value within a distribution. Z-scores standardise values to a common scale regardless of original units, enabling direct comparison across different datasets.

Measure	Formula	Interpretation
Z-Score (population)	z = (x − μ) / σ	Standard deviations above/below the population mean
Z-Score (sample)	z = (x − x̄) / s	Standard deviations above/below the sample mean
Percentile rank	PR = (# values < x) / n × 100	Percentage of data below value x
Quartile Q1	Median of lower half	25th percentile — 25% of data lies below
Quartile Q3	Median of upper half	75th percentile — 75% of data lies below
Tukey Outlier Fence	Q1 − 1.5×IQR and Q3 + 1.5×IQR	Values outside are potential outliers

Probability Rules

Probability quantifies uncertainty. All probabilities must satisfy 0 ≤ P(A) ≤ 1, and the probabilities of all possible outcomes must sum to 1. The four rules below cover the building blocks of most probability calculations.

Rule	Formula	When to use
Complement rule	P(Aᶜ) = 1 − P(A)	Finding "at least one" or "not A" scenarios
Addition rule (mutually exclusive)	P(A ∪ B) = P(A) + P(B)	Events that cannot both occur simultaneously
Addition rule (general)	P(A ∪ B) = P(A) + P(B) − P(A ∩ B)	Events that can overlap
Multiplication rule (independent)	P(A ∩ B) = P(A) × P(B)	Events where one does not affect the other
Multiplication rule (dependent)	P(A ∩ B) = P(A) × P(B\|A)	Events where one affects the probability of the other
Conditional probability	P(B\|A) = P(A ∩ B) / P(A)	Probability of B given A has already occurred
Bayes' theorem	P(A\|B) = P(B\|A) × P(A) / P(B)	Updating probability when new evidence arrives

Key Probability Distributions

Probability distributions describe how values are spread across possible outcomes. Choosing the right distribution is the foundation of correct statistical modelling.

Distribution	Formula	Mean	Variance	Use for
Normal	f(x) = (1/σ√2π) × e^−½((x−μ)/σ)²	μ	σ²	Continuous symmetric data; CLT approximation
Binomial	P(X=k) = C(n,k) × pᵏ × (1−p)^(n−k)	np	np(1−p)	Fixed n trials, success/failure outcomes
Poisson	P(X=k) = λᵏ × e^−λ / k!	λ	λ	Count of rare events in an interval
t-distribution	Heavy-tailed; df parameter	μ (df>1)	df/(df−2)	Small samples with unknown σ
Chi-square	Sum of squared standard normals	df	2×df	Categorical data tests; variance tests
F-distribution	Ratio of two chi-square variates	df₂/(df₂−2)	—	ANOVA; comparing variances

Inferential Statistics Formulas

Inferential statistics uses sample data to draw conclusions about a population. The core tools are hypothesis tests (which produce a p-value) and confidence intervals (which produce a plausible range for the parameter).

Test	Formula	df	Use for
One-sample z-test	z = (x̄ − μ₀) / (σ / √n)	—	Mean vs known μ₀, σ known
One-sample t-test	t = (x̄ − μ₀) / (s / √n)	n−1	Mean vs known μ₀, σ unknown
Two-sample Welch t	t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)	Welch–Satterthwaite approx.	Comparing two means, unequal variance
Chi-square GoF	χ² = Σ(O − E)² / E	k − 1	Observed vs expected frequencies
Chi-square independence	χ² = Σ(O − E)² / E	(r−1)(c−1)	Association between two categorical variables
CI for mean (t)	x̄ ± t* × s/√n	—	Unknown σ — uses t critical value
CI for proportion (z)	p̂ ± z* × √(p̂(1−p̂)/n)	—	Binary outcome — uses z critical value

Regression Formulas

Simple linear regression models the linear relationship between one predictor variable x and one outcome variable y. The goal is to find the line y = b₀ + b₁x that minimises the sum of squared residuals (the ordinary least squares criterion).

The slope b₁ tells you how much y changes on average for each unit increase in x. The intercept b₀ is the predicted value of y when x = 0. R² (the coefficient of determination) tells you what proportion of the variance in y is explained by x — ranging from 0 (no relationship) to 1 (perfect linear relationship).

Quantity	Formula	Interpretation
Slope	b₁ = Σ(xᵢ−x̄)(yᵢ−ȳ) / Σ(xᵢ−x̄)²	Change in y per unit increase in x
Intercept	b₀ = ȳ − b₁x̄	Predicted y when x = 0
Pearson r	r = Σ(xᵢ−x̄)(yᵢ−ȳ) / (n−1)sₓsᵧ	Linear correlation strength: −1 to +1
R-squared	R² = r²	Proportion of variance in y explained by x
Residual	e = yᵢ − ŷᵢ	Difference between observed and predicted y
RMSE	√(Σeᵢ² / (n−2))	Typical prediction error in original units

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Variance Calculator

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Correlation Calculator

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Normal Distribution Calculator

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Educational use only. Content is based on publicly documented mathematical formulas and reviewed for accuracy by the CalcMulti Editorial Team. Last updated: February 2026.