Quick Statistics Calculator
Enter a dataset for instant descriptive statistics — mean, median, mode, standard deviation, variance, and range — with full step-by-step working.
For dedicated tools see the mean, median & mode calculator, standard deviation calculator, and confidence interval calculator.
Descriptive Statistics Cheat Sheet
Descriptive statistics summarize and describe the data you actually have. They do not make predictions or generalizations to a wider population — that is the job of inferential statistics. The main categories covered by descriptive statistics are central tendency, spread, and shape.
Measures of Central Tendency
μ = population mean
Σxᵢ = sum of all values
N = population size
x̄ = sample mean
n = sample size (not population)
Median: Order values smallest to largest. If n is odd: value at position (n+1)/2. If n is even: average of values at positions n/2 and n/2 + 1.
Mode: The value appearing most often. A dataset can have no mode, one mode, or multiple modes.
Dataset: 4, 7, 7, 9, 12, 15, 21
Mean: (4+7+7+9+12+15+21) ÷ 7 = 75 ÷ 7 = 10.71
Median: 7 values → middle position = 4th → 9
Mode: 7 appears twice, all others once → 7
✓ The mean (10.71) is pulled right by the outlier 21. The median (9) is more representative here. Check with the mean, median & mode calculator.
All three agree. Report the mean — it uses every data point and is algebraically useful.
Mean is pulled right by high values. Report the median — it is resistant to extremes.
Mean is pulled left by low values. Report the median for the same reason.
If mean and median are far apart, the data is skewed or contains outliers. Report both and state which you are using as the primary measure, and why.
Individual calculators: Mean Calculator · Median Calculator · Interactive Visualizer
Measures of Spread / Variability
n−1 = degrees of freedom (unbiased estimate)
Population: σ² = Σ(xᵢ−μ)² / N
s = sample SD (denominator n−1)
σ = population SD (denominator N)
Same units as original data
σ and σ² use denominator N (you have every data point — a full population). s and s² use denominator n−1 (you have a sample). Mixing them up gives wrong standard errors in every downstream test. See standard deviation vs. variance.
| Statistic | Symbol | Formula | Use When |
|---|---|---|---|
| Population mean | μ | Σxᵢ / N | Full population (census) |
| Sample mean | x̄ | Σxᵢ / n | Symmetric sample data |
| Median | — | Middle of sorted data | Skewed data / outliers |
| Mode | — | Most frequent value | Nominal / categorical data |
| Range | — | Max − Min | Quick rough spread |
| Population variance | σ² | Σ(xᵢ−μ)² / N | Full population available |
| Sample variance | s² | Σ(xᵢ−x̄)² / (n−1) | Working with a sample |
| Population SD | σ | √σ² | Full population; same units |
| Sample SD | s | √s² | Sample data — most common |
| IQR | IQR | Q3 − Q1 | Middle 50%; outlier-resistant |
| CV | CV | (s / x̄) × 100% | Compare spread across scales |
| Five-number summary | — | Min, Q1, Median, Q3, Max | Box plots; outlier detection |
Tools: Standard Deviation Calculator · IQR Calculator · Box Plot Generator · Percentile Calculator
Probability Cheat Sheet
| Rule | Formula | Plain English |
|---|---|---|
| Addition | P(A∪B) = P(A) + P(B) − P(A∩B) | Prob. of A or B (subtract overlap) |
| Mutually exclusive | P(A∪B) = P(A) + P(B) | No overlap — just add |
| Multiplication | P(A∩B) = P(A) × P(B|A) | Prob. of A and B together |
| Independence | P(A∩B) = P(A) × P(B) | Events don't affect each other |
| Complement | P(Aᶜ) = 1 − P(A) | Prob. of NOT A |
| Conditional | P(A|B) = P(A∩B) / P(B) | Prob. of A given B already happened |
| Bayes' Theorem | P(A|B) = [P(B|A) × P(A)] / P(B) | Update belief given new evidence |
| Z-score | z = (x − μ) / σ | How many SDs from the mean |
| Distribution | Use Case | Parameters | Mean | Variance |
|---|---|---|---|---|
| Normal | Continuous data (heights, errors) | μ, σ | μ | σ² |
| Binomial | Count successes in n yes/no trials | n, p | np | np(1−p) |
| Poisson | Count of rare events per interval | λ | λ | λ |
| Exponential | Time between Poisson events | λ | 1/λ | 1/λ² |
Tools: Z-Score Calculator · Bell Curve Generator · Bayes' Theorem Calculator
Inferential Statistics Cheat Sheet
Descriptive statistics describe your sample. Inferential statistics use that sample to draw conclusions about a population you did not fully measure. The two main tools are confidence intervals (estimating a parameter) and hypothesis tests (making a binary decision about a claim).
Confidence Intervals
n = 40, x̄ = 52, s = 8
SE = 8 / √40 = 8 / 6.32 = 1.265
t* with df = 39, 95% CI → t* ≈ 2.023
ME = 2.023 × 1.265 = 2.559
CI: 52 ± 2.559 → (49.44, 54.56)
✓ We are 95% confident the population mean falls between 49.44 and 54.56. Verify with the confidence interval calculator. For proportions, use the Wilson score interval.
A 95% CI does NOT mean "there is a 95% probability the true value is inside this interval." The true value is fixed. It means: if you repeated the study many times, 95% of the intervals built that way would contain the true value. See how to interpret a confidence interval.
Hypothesis Testing Framework
Apply to any statistical test
State H₀ and H₁. H₀ = no effect / no difference. H₁ = what you are trying to detect. Example: H₀: μ = 50 vs. H₁: μ ≠ 50
Set α. Usually 0.05 — your acceptable false positive rate.
Calculate test statistic (t, z, F, χ²) from your sample.
Find p-value — probability of data this extreme if H₀ were true.
Decide: p < α → reject H₀. p ≥ α → fail to reject H₀. Never say "accept H₀."
The p-value is the probability of obtaining results at least as extreme as yours, assuming the null hypothesis is true. It is NOT the probability your hypothesis is correct. It is NOT the probability the null hypothesis is true. A small p-value means your data would be unlikely under H₀ — not that H₀ is definitely wrong.
p < 0.05 does not mean the effect is large or meaningful. With a large enough sample, even a trivial difference becomes statistically significant. Always pair a p-value with an effect size (Cohen's d, R², etc.). See what does p-value mean.
Type I error (α): Rejecting H₀ when it is true. False positive. Probability = α.
Type II error (β): Failing to reject H₀ when it is false. False negative.
Power (1−β): Probability of correctly detecting a real effect. Increases with larger n and larger effects.
Which Statistical Test Should I Use?
Use the Shapiro-Wilk test (best for n < 50) or check a histogram and Q-Q plot. The normality tests guide covers every method. Use the interactive statistical test selector for a guided walkthrough.
| Test | Use Case | Data Type | Key Assumption | Non-Parametric Alt. |
|---|---|---|---|---|
| One-sample t-test | Compare mean to known value | Continuous | Normality or n≥30 | Wilcoxon signed-rank |
| Independent t-test | Compare two group means | Continuous | Normality, independence | Mann-Whitney U |
| Paired t-test | Before/after, same subjects | Continuous | Normal differences | Wilcoxon signed-rank |
| One-way ANOVA | Compare 3+ group means | Continuous | Normality, equal variance | Kruskal-Wallis |
| Chi-square | Association between 2 cat. vars | Categorical | Expected freq ≥ 5 | Fisher's exact test |
| Pearson r | Linear relationship | Continuous | Linearity, normality | Spearman ρ |
| Linear regression | Predict continuous outcome | Continuous | Linearity, normal residuals | — |
| Logistic regression | Predict binary outcome | Binary outcome | Independence, n≥10/predictor | — |
t-Test Cheat Sheet
| Test | Formula | df |
|---|---|---|
| One-sample | t = (x̄ − μ₀) / (s / √n) | n − 1 |
| Independent | t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂) | Welch approximation |
| Paired | t = d̄ / (s_d / √n) where d = difference per pair | n − 1 |
Tables: t-Distribution Table · t-Test Calculator · t-test vs. z-test guide
Chi-Square Test Cheat Sheet
O = observed frequency
E = (row total × col total) / grand total
df = (rows−1) × (cols−1)
Two uses: test of independence (are two categorical variables associated?) and goodness of fit (does observed distribution match a theoretical one?). All expected cell frequencies must be ≥ 5 — use Fisher's exact test if not. Look up critical values in the chi-square table or use the chi-square calculator.
ANOVA Cheat Sheet
A significant F tells you at least one group mean differs — not which ones. Run Tukey's HSD post-hoc for pairwise comparisons. Tables: F-Table · Tukey's Q Table. Two-way ANOVA adds a second factor and tests interaction effects.
Correlation and Regression Cheat Sheet
| Method | Formula / Notes | When to Use |
|---|---|---|
| Pearson r | r = Σ[(xᵢ−x̄)(yᵢ−ȳ)] / [(n−1)·sₓ·s_y] | Linear, continuous, parametric |
| Spearman ρ | Pearson r applied to ranks | Ordinal or non-normal continuous |
| Kendall τ | Rank-based concordance measure | Small samples or many ties |
| Linear regression | ŷ = b₀ + b₁x | b₁ = r·(s_y/s_x) | R² = r² | Predict continuous outcome |
A high r value shows two variables move together. It says nothing about which causes which. See correlation vs. causation. Use the correlation calculator for Pearson and Spearman r.
Statistics Symbols & Notation Cheat Sheet
If you have ever stared at a statistics formula and felt confused by the Greek letters, this table is for you. Learn these once and every formula becomes readable.
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| μ | Mu | Population mean | H₀: μ = 50 |
| σ | Sigma | Population standard deviation | z = (x − μ) / σ |
| σ² | Sigma squared | Population variance | σ² = Σ(xᵢ−μ)² / N |
| x̄ | X-bar | Sample mean | x̄ = Σxᵢ / n |
| s | s | Sample standard deviation | SE = s / √n |
| s² | s squared | Sample variance | s² = Σ(xᵢ−x̄)² / (n−1) |
| n | n | Sample size | df = n − 1 |
| N | N | Population size | μ = Σx / N |
| p | p | Population proportion | H₀: p = 0.5 |
| p̂ | p-hat | Sample proportion | p̂ = x / n |
| ρ | Rho | Population correlation / Spearman corr. | H₀: ρ = 0 |
| r | r | Sample Pearson correlation | r = 0.82 |
| α | Alpha | Significance level / Type I error | α = 0.05 |
| β | Beta | Type II error / regression coefficient | Power = 1−β |
| H₀ | H-naught | Null hypothesis | H₀: μ₁ = μ₂ |
| H₁ | H-one | Alternative hypothesis | H₁: μ₁ ≠ μ₂ |
| df | Degrees of freedom | Values free to vary in calculation | df = n−1 |
| SE | Standard error | SD of the sampling distribution | SE = s / √n |
| z | z-score | Standard normal test statistic | z = (x̄−μ₀) / (σ/√n) |
| t | t-statistic | t-distribution test statistic | t = (x̄−μ₀) / (s/√n) |
| F | F-statistic | Ratio of variances (ANOVA) | F = MS_between / MS_within |
| χ² | Chi-squared | Categorical data test statistic | χ² = Σ(O−E)²/E |
| Σ | Sigma (cap.) | Summation — add up all values | x̄ = Σxᵢ / n |
| ε | Epsilon | Error term in regression | y = b₀ + b₁x + ε |
| R² | R-squared | Proportion of variance explained | R² = 0.85 → 85% explained |
| b₀ | b-naught | Regression y-intercept | ŷ = b₀ + b₁x |
| b₁ | b-one | Regression slope | b₁ = 2.3 → y rises 2.3 per unit x |
AP Statistics Cheat Sheet
RNI Conditions Checklist
Before any inference procedure on the AP exam, check all three conditions and write them out explicitly. Partial credit is lost when students skip this.
- Random (R): Data came from a random sample or a randomized experiment — allows generalization to the population.
- Normal (N): For means: n ≥ 30 (CLT) or population stated as normal. For proportions: np ≥ 10 AND n(1−p) ≥ 10.
- Independent (I): 10% condition — n ≤ 10% of population when sampling without replacement.
10 Common AP Stats Mistakes
"p = 0.03 means there is a 97% chance H₀ is false" — wrong. p-value is the probability of your data given H₀ is true, not the probability of the hypothesis.
Skipping Random, Normal, and Independent costs exam points every time. State each condition with justification from context.
x̄ and s are sample statistics. μ and σ are population parameters. The distinction matters for every test statement.
House prices, incomes, and response times are typically right-skewed. Report the median unless the question specifies otherwise.
"We are 95% confident that μ falls between 47 and 53" must name what μ represents: "...the mean exam score for all students in the district."
Statistics never proves anything. Write "there is sufficient evidence to conclude..." not "this proves that..."
Failing to reject H₀ means your data is consistent with it — not that it is true. Write "fail to reject H₀."
A two-sided p-value is double the one-sided value. Make sure your stated H₁ matches the direction of the test you calculated.
SE = s/√n is the SD of the sampling distribution — not the SD of the original data. These are completely different quantities.
If data came from volunteers or a convenience sample, you cannot generalize to a broader population even if the test is significant.
Statistics Cheat Sheet for Data Science
Key Concepts for ML
Bias-variance tradeoff: Total error = bias² + variance + irreducible noise. High bias = underfitting (model too simple). High variance = overfitting (fits training data, fails on new data). Cross-validation estimates generalization error.
| Metric | Formula | Use When |
|---|---|---|
| Accuracy | (TP + TN) / Total | Balanced classes |
| Precision | TP / (TP + FP) | False positives are costly |
| Recall | TP / (TP + FN) | False negatives are costly |
| F1 Score | 2 × (Precision × Recall) / (Precision + Recall) | Imbalanced classes |
| MSE | Σ(yᵢ − ŷᵢ)² / n | Regression; penalizes large errors |
| RMSE | √MSE | Same units as target variable |
| MAE | Σ|yᵢ − ŷᵢ| / n | Robust to outliers |
Statistics in Python
import numpy as np from scipy import stats # Descriptive statistics np.mean(x) # sample mean np.std(x, ddof=1) # sample SD (ddof=1 for n-1) np.var(x, ddof=1) # sample variance # t-tests stats.ttest_1samp(x, popmean=10) # one-sample stats.ttest_ind(x1, x2) # independent stats.ttest_rel(x1, x2) # paired # Chi-square stats.chi2_contingency(table) # chi-sq + expected # Correlation stats.pearsonr(x, y) # Pearson r + p stats.spearmanr(x, y) # Spearman rho + p # Linear regression stats.linregress(x, y) # slope, intercept, r, p, se
Statistics in Excel
| Task | Excel Function |
|---|---|
| Sample mean | AVERAGE(range) |
| Sample SD | STDEV.S(range) |
| Population SD | STDEV.P(range) |
| Sample variance | VAR.S(range) |
| Population variance | VAR.P(range) |
| Median | MEDIAN(range) |
| Mode | MODE.SNGL(range) |
| t-test p-value | T.TEST(array1, array2, tails, type) |
| Chi-square test | CHISQ.TEST(actual_range, expected_range) |
| Pearson correlation | CORREL(array1, array2) |
| Regression slope | SLOPE(known_ys, known_xs) |
| Regression intercept | INTERCEPT(known_ys, known_xs) |
| Z-score | STANDARDIZE(x, mean, std_dev) |
Statistics for Business & Research
For most business questions, the workflow is: describe the data first (means, ranges, charts), then test whether observed differences are real or just noise (hypothesis tests), then quantify how big the effect is (effect sizes, R²). The business statistics guide applies this to common real-world scenarios.
A/B Testing
Comparing two page variants, emails, or product features. Use an independent t-test (continuous outcome) or z-test for proportions (conversion rate).
t-test / z-testSalary / Income Analysis
Income data is almost always right-skewed. Report median salary, not mean. Use IQR instead of standard deviation for spread.
Median + IQRSurvey Analysis
Likert-scale items are ordinal — use Spearman correlation and median. Nominal category responses — use chi-square for associations.
Median / Chi-sqInvestment Returns
For compound returns and growth rates, the geometric mean is the correct average — not the arithmetic mean.
Geometric MeanQuality Control
Use control charts (mean ± 3σ) to flag process variation. ANOVA compares means across production lines or time periods.
Mean + ANOVAClinical Trials
Paired t-test for pre/post measurements on the same patients. Report both statistical significance (p-value) and clinical significance (effect size).
Paired t-testCommon Statistical Pitfalls
A trend that appears in combined data can reverse or disappear when the data is separated into groups. A confounding variable (group size, demographic mix) drives the reversal.
Analyzing only the observations that made it through a filter — companies that survived, students who graduated — and ignoring the failures leads to wrong conclusions about what predicts success.
Running 20 hypothesis tests at α = 0.05 gives roughly a 64% chance of at least one false positive, even if no real effects exist. Apply Bonferroni correction or control the false discovery rate.
A hidden variable correlated with both predictor and outcome creates a spurious association. Observational studies cannot control for unmeasured confounders the way randomized experiments can.
When the sample is not representative of the population, every inference from it is suspect — regardless of sample size. Self-selected samples (opt-in surveys, online reviews) are especially prone to this.
A correlation found at the group level (countries, schools, cities) does not automatically apply to individuals within those groups.
Key Takeaways
Descriptive statistics summarize your sample. Inferential statistics use that sample to make claims about a population.
Use the mean for symmetric continuous data. Use the median when data is skewed or contains outliers.
σ and σ² are for populations (denominator N). s and s² are for samples (denominator n−1). Never mix them.
The p-value is the probability of your data given H₀ is true — not the probability your hypothesis is correct.
Statistical significance (p < 0.05) does not equal practical importance. Always report effect size alongside p-values.
A 95% confidence interval does not mean 95% probability the true value is inside it — it means 95% of such intervals contain the true value over repeated sampling.
Correlation does not imply causation. High r only shows variables move together.
Check normality before choosing between parametric and non-parametric tests — Shapiro-Wilk test or histogram + Q-Q plot.
For ANOVA with a significant F, run Tukey's HSD post-hoc to identify which specific groups differ.
FAQ — Statistics Cheat Sheet
A complete stats cheat sheet needs descriptive formulas (mean, median, mode, variance, SD, IQR), probability rules (addition, multiplication, conditional, Bayes), inferential tools (confidence intervals, t-tests, chi-square, ANOVA, regression), a test-selection guide, and a symbols table. The most useful ones pair every formula with plain-English interpretation and a worked example — which is exactly what this page does.
The standard error formula — SE = s / √n — has the broadest impact. It appears in every confidence interval, every t-test, and every sampling distribution calculation. Understanding why dividing by √n matters (larger samples produce more precise estimates) unlocks almost all of inferential statistics.
Start with your outcome variable type. Continuous outcome with groups to compare → t-tests or ANOVA. Two continuous variables to relate → correlation or regression. Categorical outcome → chi-square or logistic regression. Then check normality (Shapiro-Wilk, histogram). If the normality assumption fails, switch to the non-parametric equivalent. The decision tree in the inferential section above walks through every branch.
It means that if the null hypothesis were true, there would be a 5% chance of observing results as extreme as yours by chance alone. It is not the probability that your hypothesis is correct, and it tells you nothing about the size or practical importance of the effect. Whether 0.05 is "significant" depends on your pre-specified alpha level.
Variance is the average squared deviation from the mean. Standard deviation is its square root — which returns the measure to the original units of the data, making it interpretable. If your data is in kilograms, the standard deviation is in kilograms; the variance is in kilograms squared, which has no intuitive meaning. Full comparison: standard deviation vs. variance.
Descriptive statistics summarize the data you collected. Inferential statistics use that sample to make claims about a larger population — via hypothesis tests and confidence intervals. If you calculate the average age of 200 survey respondents, that is descriptive. If you use that average to estimate the average age of the entire country with a margin of error, that is inferential. Full article: descriptive vs. inferential statistics.
For a 95% CI for a mean: (1) Calculate x̄ and s. (2) SE = s / √n. (3) Find t* with df = n−1 (or z* = 1.96 if σ known and n large). (4) ME = t* × SE. (5) CI = x̄ ± ME. Use the confidence interval calculator to check your work instantly.
This page is a learning and reference tool. For exams allowing reference sheets, download the free printable PDF below — it fits the key formulas, test-selection decision tree, and symbols table on two print-ready pages. For AP Statistics: the College Board provides its own formula sheet at the exam; this page helps you understand and practice with those formulas before exam day.
📥 Free Printable Statistics Cheat Sheet PDF
Download the complete statistics formula reference PDF — all formulas, the test-selection decision tree, the symbols table, and the hypothesis testing framework on 2 print-ready A4 pages. Every formula on statisticsfundamentals.com has a dedicated calculator to check your work instantly.
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