Tools in This Collection
Standard Deviation Calculator
Calculate mean, variance, and standard deviation for population or sample data
Probability Calculator
Compute probabilities for binomial, normal, and Poisson distributions
Permutation and Combination Calculator
Calculate arrangements (order matters) and selections (order doesn't matter)
Z-Score Calculator
Convert between raw scores and z-scores with normal distribution probabilities
Confidence Interval Calculator
Build confidence intervals at 90%, 95%, or 99% confidence levels
Chi-Square Calculator
Test independence between categorical variables with chi-square analysis
T-Test Calculator
Compare group means with one-sample or independent samples t-tests
Correlation Calculator
Compute Pearson correlation coefficient and test significance of relationships
Statistics and Probability Workflow
Statistical analysis follows a natural progression: describe your data, then make inferences about the broader population. Start with descriptive statistics to understand what you have, then apply inferential tests to determine what conclusions you can draw.
Begin with the Standard Deviation Calculator to measure spread in your dataset. Standard deviation tells you how tightly clustered values are around the mean. A dataset {10, 10, 10, 10} has SD = 0. A dataset {2, 8, 14, 20} has mean = 11 and SD ≈ 6.7. The calculator computes both population SD (divide by N) and sample SD (divide by N-1 — use this when your data is a sample from a larger population).
For z-scores, a value's z-score tells you how many standard deviations it falls from the mean: z = (x - μ) / σ. A test score of 85 in a class where μ = 75 and σ = 10 gives z = (85 - 75) / 10 = 1.0 — one standard deviation above average. The Z-Score Calculator converts between raw scores and z-scores and looks up normal distribution probabilities. A z-score of 1.96 corresponds to the 97.5th percentile — the boundary for 95% confidence intervals.
For hypothesis testing, the T-Test Calculator compares means between groups. Use a one-sample t-test to compare a sample mean to a known value, or an independent samples t-test to compare two groups. The result is a t-statistic and p-value — if p < 0.05 (your significance threshold), the difference is statistically significant. The Chi-Square Calculator tests independence between categorical variables (e.g., does product preference differ by gender?). The Confidence Interval Calculator constructs intervals showing the range where the true population parameter likely falls with 90%, 95%, or 99% confidence.
For relationships between variables, the Correlation Calculator computes the Pearson correlation coefficient (r), ranging from -1 (perfect negative) to +1 (perfect positive). r = 0.8 indicates a strong positive relationship, r = 0 indicates no linear relationship. The Permutation and Combination Calculator handles counting problems: choosing 3 people from 10 for a committee (order doesn't matter) = C(10,3) = 120. Arranging 3 books on a shelf from 10 options (order matters) = P(10,3) = 720. The Probability Calculator covers binomial, normal, and Poisson distributions.
Frequently Asked Questions
What is standard deviation and how do I interpret it?
Standard deviation (SD) measures how spread out values are around the mean. A small SD means values cluster tightly; a large SD means values are spread widely. For a normal distribution, about 68% of values fall within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. If exam scores have mean 75 and SD 10, about 68% of scores fall between 65 and 85. Use population SD (divide by N) for complete datasets, sample SD (divide by N-1) for samples.
When should I use a t-test vs chi-square test?
Use a t-test when comparing means of a continuous numeric variable between groups. For example: do men and women differ in average height? Do students taught with method A score higher than method B? Use a chi-square test when testing whether two categorical variables are related. For example: is smoking status (yes/no) related to cancer diagnosis (yes/no)? Is product preference (A/B/C) related to customer age group (young/middle/senior)?
What is the difference between correlation and causation?
Correlation measures how strongly two variables move together. Ice cream sales and drowning rates are positively correlated (r ≈ 0.8) — but neither causes the other. Both are caused by hot weather. Correlation only shows linear relationship strength; it says nothing about cause and effect. Establishing causation requires controlled experiments or careful causal inference methods that rule out confounding variables. A high correlation (r > 0.7) suggests a relationship worth investigating, but doesn't prove causation.
What does a p-value of 0.05 mean?
A p-value of 0.05 means there is a 5% probability of seeing results as extreme as yours (or more extreme) if there were actually no real effect — just random chance. The 0.05 threshold is a convention, not a law. If p < 0.05, you reject the null hypothesis and call the result statistically significant. But statistical significance doesn't mean practical significance — a very large sample can make tiny, meaningless differences statistically significant.