What is the formula for variance vs standard deviation?

Variance is the average of the squared differences from the mean: for a population, σ² = Σ(x − μ)² / N; for a sample, s² = Σ(x − x̄)² / (n−1). Standard deviation is simply the square root of the variance: σ = √(σ²) for a population and s = √(s²) for a sample. Standard deviation is in the same units as the original data, which makes it more interpretable.

What is the difference between population and sample standard deviation?

Population standard deviation (σ) divides by N and is used when you have data for the entire population. Sample standard deviation (s) divides by n−1 (Bessel's correction) and is used when your data is a sample from a larger population. The n−1 correction produces an unbiased estimate of the true population variability.

How do I calculate a z-score step by step?

To calculate a z-score: (1) identify the data value x, the population mean μ, and the population standard deviation σ; (2) subtract the mean from the value: x − μ; (3) divide by the standard deviation: (x − μ) / σ. The result tells you how many standard deviations above or below the mean that value is.

What formulas are on the AP Statistics formula sheet?

The College Board provides an AP Statistics formula sheet during the exam. It includes formulas for the sample mean, sample standard deviation, standardized test statistic, confidence interval, chi-square statistic, and key probability distributions. This cheat sheet covers all those formulas plus the probability rules, combinations, and permutations that appear throughout the course.

What is the difference between combinations and permutations?

Permutations count arrangements where order matters: nPr = n! / (n−r)!. Combinations count selections where order does not matter: nCr = n! / (r!(n−r)!). For the same n and r, there are always fewer combinations than permutations because combinations treat different orderings of the same group as one selection.

What is the formula for a confidence interval?

Every confidence interval follows the same form: estimate ± (critical value) × (standard error). For a proportion, this is p-hat ± z* times the square root of p-hat(1 − p-hat)/n. For a mean, this is x-bar ± t* times s/√n, using a t-distribution with n−1 degrees of freedom since the population standard deviation is unknown.

How do you calculate a chi-square test statistic?

The chi-square statistic is the sum, across every cell in a two-way table, of (observed − expected)² divided by expected, where expected count equals row total times column total divided by grand total. As of the AP Statistics course effective the 2026-27 school year, this formula is used only for the test for homogeneity and the test for independence — the chi-square goodness-of-fit test was removed.

What is the formula for the least-squares regression line?

The least-squares regression line is y-hat = b0 + b1x, where the slope b1 = r × (sy/sx) and the intercept b0 = y-bar − b1 × x-bar. Here r is the correlation coefficient, sy and sx are the standard deviations of y and x, and x-bar and y-bar are their means.

Statistics Formulas Cheat Sheet: Every Formula You Need with Examples

This statistics formulas cheat sheet is a single-page reference covering measures of center, measures of spread, z-scores, probability rules, combinations and permutations, the normal distribution, confidence intervals, hypothesis testing, chi-square tests, and correlation and regression. It is designed for AP Statistics students, introductory college statistics courses, and anyone who needs a fast, accurate formula lookup.

Every formula on this page is rendered as crawlable HTML text — not locked in an image or PDF — and links directly to a free step-by-step calculator so you can verify each formula in action. All formulas verified against the College Board AP Statistics formula sheet and the AP Statistics Course and Exam Description effective Fall 2026. Last reviewed September 2026.

Reflects the 2027 AP Statistics exam. The course was redesigned effective the 2026–27 school year. If you are studying the inference or chi-square sections below, see what changed for 2027 and which topics were removed.

Statistics Symbol Key: Population vs Sample

One of the most common sources of errors in statistics is confusing population and sample notation. This table is your reference before touching any formula.

Symbol	Meaning	Population or sample?
μ (mu)	Population mean	Population
x̄ (x-bar)	Sample mean	Sample
σ (sigma)	Population standard deviation	Population
s	Sample standard deviation	Sample
σ² (sigma-squared)	Population variance	Population
s²	Sample variance	Sample
N	Population size (total)	Population
n	Sample size	Sample
P	Population proportion	Population
p̂ (p-hat)	Sample proportion	Sample
Σ	Sum of all values	Both
ŷ	Predicted value (regression)	Sample
p̂₁, p̂₂	Sample proportions for two groups	Sample
x̄₁, x̄₂	Sample means for two groups	Sample
s₁, s₂	Sample standard deviations for two groups	Sample
t	Test statistic used when σ is unknown	Sample
df	Degrees of freedom	Sample
χ² (chi-square)	Chi-square test statistic	Sample
H₀, H_a	Null and alternative hypothesis	—
α (alpha)	Significance level (commonly 0.05)	—
r	Correlation coefficient	Sample
b₀, b₁	Regression intercept and slope	Sample

Measures of Center Formulas

Sample Mean

x̄ = Σx / n

Σx = sum of all data values, n = number of values in the sample.

Example: Data set {4, 7, 7, 9, 13} → x̄ = (4+7+7+9+13)/5 = 40/5 = 8

Median

The median is the middle value when data is ordered from least to greatest.

Odd number of values: Median = middle value at position (n+1)/2
Even number of values: Median = average of the two middle values at positions n/2 and n/2+1

Example: {4, 7, 7, 9, 13} → Median = 7 (3rd value of 5)

Mode

The mode is the value that appears most frequently. No formula — just count. A data set may have no mode, one mode, or multiple modes.

Example: {4, 7, 7, 9, 13} → Mode = 7 (appears twice)

Weighted Mean

x̄_w = Σ(w · x) / Σw

w = weight for each value, x = value. Used for grade point averages, where each course credit count differs.

Use statistics formulas for center together: mean for symmetric data, median when outliers are present, mode for categorical data.

Calculate mean, median & mode instantly with the MathInSite calculator — shows all three in one step.

Measures of Spread (Dispersion) Formulas

Measure	Formula	Best used when…
Range	max − min	Quick, rough estimate; sensitive to outliers
IQR	Q3 − Q1	Data has outliers; more robust than range
Population variance	σ² = Σ(x − μ)² / N	You have complete population data
Sample variance	s² = Σ(x − x̄)² / (n−1)	You have sample data (most common)
Population SD	σ = √(σ²)	Complete population data
Sample SD	s = √(s²)	Sample data (most exams and real-world use)
Coefficient of Variation	CV = (s / x̄) × 100%	Comparing spread across different scales or units

Why Sample Variance Divides by n−1

Dividing by n−1 instead of n is called Bessel's correction. A sample tends to underestimate the true population spread when you divide by n. The correction adjusts for this bias and produces an unbiased estimate of population variance. In practice: always use n−1 (sample variance s²) unless you explicitly have data for the entire population.

Standard Deviation in Plain English

Standard deviation measures the average distance each value is from the mean. A low SD means data is tightly clustered; a high SD means data is spread out. Because it is the square root of variance, SD is in the same units as the original data — making it far more interpretable than raw variance.

Common mistake: Using population SD (σ, divides by N) when you have sample data. Always default to sample SD (s, divides by n−1) unless explicitly told you have the full population.

Check your standard deviation calculation with the step-by-step standard deviation calculator.

Z-Score Formula

z = (x − μ) / σ

z = z-score (number of standard deviations from the mean), x = data value, μ = population mean, σ = population standard deviation.

The z-score tells you how many standard deviations above or below the mean a particular value falls. A z-score of +2 means the value is 2 standard deviations above average; −1.5 means 1.5 standard deviations below.

Standardizing a Sample Mean

z = (x̄ − μ) / (σ / √n)

Used when working with sample means instead of individual values. The denominator σ/√n is the standard error.

Worked Example

A student scores 82 on a test where the class mean μ = 75 and standard deviation σ = 7.

z = (82 − 75) / 7
z = 7 / 7
z = 1.0

The student scored exactly 1 standard deviation above the mean.

Z-Score Interpretation Guide

Z-score range	Interpretation
z < −2	Unusually low — below 97.7% of the distribution
−2 ≤ z ≤ −1	Below average
−1 < z < 1	Typical range (about 68% of data)
1 ≤ z ≤ 2	Above average
z > 2	Unusually high — above 97.7% of the distribution

Try the z-score calculator — enter a value, mean, and SD to get the z-score and its percentile instantly.

Probability Formulas

Rule	Formula	When to use
Basic probability	P(A) = favorable / total	All outcomes equally likely
Complement rule	P(A') = 1 − P(A)	Easier to find what doesn't happen
Addition rule (general)	P(A or B) = P(A) + P(B) − P(A and B)	A and B can both occur
Addition rule (mutually exclusive)	P(A or B) = P(A) + P(B)	A and B cannot both occur
Multiplication rule (general)	P(A and B) = P(A) × P(B\|A)	A and B are dependent events
Multiplication rule (independent)	P(A and B) = P(A) × P(B)	A and B do not affect each other
Conditional probability	P(B\|A) = P(A and B) / P(A)	Probability of B given A already happened

Complement rule shortcut: If a problem asks for "at least one" of something, it is almost always easier to calculate P(none) and subtract from 1: P(at least one) = 1 − P(none).

Combinations and Permutations Formulas

	Permutations (order matters)	Combinations (order does not matter)
Formula	nPr = n! / (n − r)!	nCr = n! / (r!(n − r)!)
Also written	P(n, r)	C(n, r) or (n choose r)
Example	Race finishes — 1st, 2nd, 3rd matter	Committee selection — only who is chosen

Permutations

nPr = n! / (n − r)!

Count the number of ways to arrange r items from n total, where the order of arrangement matters.

Example: How many ways can 3 runners finish 1st, 2nd, and 3rd from a field of 8? → 8P3 = 8! / (8−3)! = 8!/5! = 8×7×6 = 336

Combinations

nCr = n! / (r!(n − r)!)

Count the number of ways to select r items from n total, where the order of selection does not matter.

Example: How many ways can a 3-person committee be chosen from 8 candidates? → 8C3 = 8! / (3!×5!) = 336/6 = 56

Factorial Quick Reference

0! = 1 (by definition) · 1! = 1 · 2! = 2 · 3! = 6 · 4! = 24 · 5! = 120 · 6! = 720 · 7! = 5,040 · 8! = 40,320

Common mistake: Forgetting that 0! = 1. This comes up in combination formulas when r = 0 or r = n.

Use the combinations & permutations calculator to count arrangements and selections with step-by-step working.

Normal Distribution Formulas and the Empirical Rule

Standard Normal Distribution

Any normal distribution can be converted to the standard normal distribution (μ = 0, σ = 1) using the z-score formula. Once standardized, you can use a z-table or calculator to find exact probabilities.

The Empirical Rule (68-95-99.7 Rule)

For any approximately normal distribution:

Range	Approx. % of data
μ ± 1σ (within 1 SD of mean)	≈ 68%
μ ± 2σ (within 2 SDs of mean)	≈ 95%
μ ± 3σ (within 3 SDs of mean)	≈ 99.7%

Important: The Empirical Rule only applies to distributions that are approximately bell-shaped (normal). Do not apply it to skewed distributions.

Using Z-Tables

A z-table gives the area (probability) to the left of a z-score under the standard normal curve. To find the probability of a range: (1) convert each bound to a z-score, (2) look up the area in the z-table, (3) subtract areas if needed. The z-score calculator handles this automatically — no table needed.

Confidence Interval Formulas

A confidence interval estimates a population parameter using a range built from sample data. Every confidence interval follows the same general form:

estimate ± (critical value) × (standard error)

One-Sample Interval for a Proportion

p̂ ± z* √(p̂(1 − p̂) / n)

Requires at least 10 successes and 10 failures in the sample (the large counts condition).

One-Sample Interval for a Mean

x̄ ± t* (s / √n), df = n − 1

Uses a t-distribution with n − 1 degrees of freedom, because the population standard deviation σ is unknown and is being estimated by the sample standard deviation s.

Two-Sample Interval for a Difference in Proportions

(p̂₁ − p̂₂) ± z* √(p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂)

Two-Sample Interval for a Difference in Means

(x̄₁ − x̄₂) ± t* √(s₁²/n₁ + s₂²/n₂)

Use this when: you want a range of plausible values for a parameter rather than a single yes/no test decision. A confidence interval and a two-sided hypothesis test at the same significance level always agree on whether to reject a claim.

Build any of these intervals instantly with the confidence interval & hypothesis test calculator — enter your sample statistics and get the interval, the work, and the interpretation. Or practice writing one up with the AP Statistics FRQ trainer.

Hypothesis Testing Formulas

Every hypothesis test statistic has the same shape: how far the sample statistic is from the claimed parameter, measured in standard errors.

test statistic = (statistic − parameter) / (standard error of the statistic)

One-Sample Test for a Proportion

z = (p̂ − p₀) / √(p₀(1 − p₀) / n)

p₀ = the claimed (hypothesized) population proportion.

One-Sample Test for a Mean

t = (x̄ − μ₀) / (s / √n), df = n − 1

μ₀ = the claimed (hypothesized) population mean.

Two-Sample Test for a Difference in Proportions

Uses a pooled proportion, since H₀ assumes the two population proportions are equal:

p̂_c = (x₁ + x₂) / (n₁ + n₂)

z = (p̂₁ − p̂₂) / √(p̂_c(1−p̂_c)(1/n₁ + 1/n₂))

Two-Sample Test for a Difference in Means

t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Reading a P-Value

The p-value is the probability of getting a result at least as extreme as the sample, assuming the null hypothesis is true.

p-value ≤ α (commonly 0.05): reject H₀ — convincing evidence for H_a
p-value > α: fail to reject H₀ — not enough evidence

Common mistake: "Fail to reject H₀" does not mean H₀ is proven true — only that the sample did not provide convincing evidence against it.

Run a hypothesis test step by step with the confidence interval & hypothesis test calculator — z-tests and t-tests for one or two samples, with the test statistic, p-value, and decision shown.

Chi-Square Test Formulas

χ² = Σ (observed − expected)² / expected

Chi-square tests compare observed counts in a two-way table to the counts expected under a claim. Expected count = (row total × column total) / grand total.

Test	Compares	Degrees of freedom
Homogeneity	One categorical variable across two or more separate samples	(rows − 1)(columns − 1)
Independence	Two categorical variables measured on one sample	(rows − 1)(columns − 1)

2027 exam update: The chi-square goodness-of-fit test (for a single categorical variable against a claimed distribution) was removed from AP Statistics effective the 2026–27 school year. Every chi-square question on the current exam uses a two-way table. See the full chi-square changes breakdown for detail on what changed and why.

Correlation and Regression Formulas

Correlation Coefficient

r ranges from −1 to 1 and measures the strength and direction of a linear relationship between two quantitative variables. Values near ±1 indicate a strong linear relationship; values near 0 indicate a weak one. r is unitless and does not depend on which variable is x and which is y.

Least-Squares Regression Line

ŷ = b₀ + b₁x

Coefficient	Formula	Meaning
Slope (b₁)	b₁ = r × (s_y / s_x)	Predicted change in y for each 1-unit increase in x
Intercept (b₀)	b₀ = ȳ − b₁x̄	Predicted y when x = 0

Residuals and r²

residual = y − ŷ (observed − predicted)

r² (the coefficient of determination) is the proportion of the variation in y explained by the linear model. r² = 0.81 means the regression line explains 81% of the variation in y; the remaining 19% is unexplained by x.

Use this when: you have paired quantitative data and want to describe or predict a linear relationship. Regression is now its own unit on the AP Statistics exam (10–20% of the score), with the inference-for-slopes content removed. See what happened to regression inference for 2027.

Find the regression line, r, and r² instantly with the linear regression calculator — enter your paired data and get the equation, a scatter plot, and step-by-step work.

Quick-Reference Summary Table

Formula name	Formula
Sample mean	x̄ = Σx / n
Weighted mean	x̄w = Σ(w·x) / Σw
Range	max − min
IQR	Q3 − Q1
Population variance	σ² = Σ(x−μ)² / N
Sample variance	s² = Σ(x−x̄)² / (n−1)
Population SD	σ = √(σ²)
Sample SD	s = √(s²)
Coefficient of Variation	CV = (s/x̄) × 100%
Z-score	z = (x − μ) / σ
Z-score (sample mean)	z = (x̄ − μ) / (σ/√n)
Basic probability	P(A) = favorable / total
Complement rule	P(A') = 1 − P(A)
Addition rule	P(A or B) = P(A) + P(B) − P(A and B)
Multiplication rule	P(A and B) = P(A) × P(B\|A)
Conditional probability	P(B\|A) = P(A and B) / P(A)
Permutations	nPr = n! / (n−r)!
Combinations	nCr = n! / (r!(n−r)!)
CI for a proportion	p̂ ± z*√(p̂(1−p̂)/n)
CI for a mean	x̄ ± t*(s/√n)
Test statistic for a proportion	z = (p̂−p₀)/√(p₀(1−p₀)/n)
Test statistic for a mean	t = (x̄−μ₀)/(s/√n)
Chi-square statistic	χ² = Σ(observed−expected)²/expected
Least-squares line	ŷ = b₀ + b₁x
Regression slope	b₁ = r × (s_y/s_x)

Need a calculator for the exam? The TI-84 Plus CE is the graphing calculator most commonly approved for the AP Statistics exam. View on Amazon →
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Frequently Asked Questions

What is the formula for variance vs standard deviation?

Variance is the average of the squared differences from the mean: for a population, σ² = Σ(x−μ)² / N; for a sample, s² = Σ(x−x̄)² / (n−1). Standard deviation is the square root of variance: σ = √(σ²) for a population and s = √(s²) for a sample. Standard deviation is in the same units as the original data, making it more interpretable.
What is the difference between population and sample standard deviation?

Population standard deviation (σ) divides by N and is used when you have data for the entire population. Sample standard deviation (s) divides by n−1 (Bessel's correction) and is used when your data is a sample from a larger population. The n−1 correction produces an unbiased estimate of the true population variability.
How do I calculate a z-score step by step?

To calculate a z-score: (1) identify the data value x, the population mean μ, and the population standard deviation σ; (2) subtract the mean: x − μ; (3) divide by the standard deviation: (x − μ) / σ. The result tells you how many standard deviations above or below the mean that value falls.
What formulas are on the AP Statistics formula sheet?

The College Board AP Statistics formula sheet includes the sample mean, sample standard deviation, standardized test statistic, confidence interval, chi-square statistic, and key probability distribution formulas. This cheat sheet covers all those formulas plus the probability rules, combinations, and permutations that appear throughout the AP Statistics course.
What is the difference between combinations and permutations?

Permutations count arrangements where order matters: nPr = n! / (n−r)!. Combinations count selections where order does not matter: nCr = n! / (r!(n−r)!). For the same values of n and r, there are always fewer combinations than permutations because different orderings of the same group count as one combination.
What is the formula for a confidence interval?

Every confidence interval follows the same form: estimate ± (critical value) × (standard error). For a proportion, this is p̂ ± z* √(p̂(1 − p̂)/n). For a mean, this is x̄ ± t*(s/√n), using a t-distribution with n − 1 degrees of freedom since the population standard deviation is unknown.
How do you calculate a chi-square test statistic?

The chi-square statistic is χ² = Σ(observed − expected)² / expected, summed across every cell in a two-way table, where expected count = (row total × column total) / grand total. As of the AP Statistics course effective the 2026–27 school year, this formula is used only for the test for homogeneity and the test for independence — the chi-square goodness-of-fit test was removed.
What is the formula for the least-squares regression line?

The least-squares regression line is ŷ = b₀ + b₁x, where the slope b₁ = r × (s_y/s_x) and the intercept b₀ = ȳ − b₁x̄. Here r is the correlation coefficient, s_y and s_x are the standard deviations of y and x, and x̄ and ȳ are their means.

Every descriptive formula on this cheat sheet has a free calculator to back it up — mean, median & mode, standard deviation, z-score, and combinations & permutations. For the inference formulas, use the confidence interval & hypothesis test calculator, and for regression, the linear regression calculator. Studying for the redesigned exam? Try the AP Statistics score calculator (2027 format), work through timed FRQ practice with a self-scoring rubric, or see exactly what changed for 2027. For the calculator side, the AP Stats Desmos and TI-84 guide shows every procedure on both, and the one-page cheat sheet prints on a single sheet. Explore all free math calculators at MathInSite.

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