Statistics Calculator — Descriptive Stats, Distributions & Tests

Descriptive Statistics Explained

Descriptive statistics summarize a dataset with a handful of numbers. The mean (arithmetic average) adds every value and divides by the count, giving you the balance point of the data. The median is the middle value when data is sorted; it resists the pull of outliers that can distort the mean. The mode is the most frequently occurring value and is the only measure of central tendency that works with categorical data. Knowing when to report each measure is crucial: income data is almost always summarized with the median because a few extremely high earners skew the mean upward, while exam scores are often reported as means because they tend to follow a symmetric distribution.

Understanding Standard Deviation

Standard deviation quantifies the typical distance of data points from the mean. A small standard deviation means values cluster tightly; a large one means they are spread out. For data that follows a normal distribution, the 68-95-99.7 rule (or empirical rule) provides a quick mental model: roughly 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. Variance is simply the square of the standard deviation and has the advantage of being additive across independent variables, which is why it appears so often in theoretical work despite being harder to interpret directly in the units of your data.

Probability Distributions Guide

A normal distribution is the classic bell curve defined by its mean and standard deviation. It arises naturally whenever many small, independent effects combine, which is why heights, measurement errors, and test scores often follow it. The binomial distribution counts the number of successes in a fixed number of independent yes-or-no trials, such as coin flips, defective-item checks, or email open rates. The Poisson distribution models the number of rare events occurring in a fixed interval of time or space — think website hits per minute, accidents per month, or typos per page. Choosing the right distribution is the foundation of every probability calculation and hypothesis test.

Hypothesis Testing Step by Step

Hypothesis testing follows a five-step process. First, state the null hypothesis (H₀), which represents the status quo or no effect, and the alternative hypothesis (H₁), which is what you hope to demonstrate. Second, choose a significance level (α), most commonly 0.05. Third, calculate the test statistic from your sample data. Fourth, find the p-value, which is the probability of obtaining results at least as extreme as yours if H₀ were true. Fifth, compare the p-value to α: if p ≤ α, reject H₀ in favor of H₁; otherwise, fail to reject H₀. Failing to reject does not prove H₀ is true — it simply means your data did not provide sufficient evidence against it.

Linear Regression Basics

Linear regression finds the straight line that best fits a scatter plot of X-Y data. The line equation ŷ = mx + b minimizes the sum of squared residuals (the vertical distances between each data point and the line). The R² value, or coefficient of determination, tells you what fraction of the variance in Y is explained by X. An R² of 0.85 means X accounts for 85% of the variation in Y. However, a high R² does not imply causation — two variables can be strongly correlated because they share a common cause, or purely by coincidence (spurious correlation).

Which Statistical Test Should I Use?

Choosing the right test depends on your data and research question. Use a one-sample t-test when you have one group and want to compare its mean to a known value. Use a two-sample t-test to compare the means of two independent groups. Use a z-test for proportions when comparing percentages or rates between two groups (e.g., conversion rates in an A/B test). For correlation and prediction with continuous variables, use regression. When your data is not normally distributed or you have small samples, consider non-parametric alternatives. This calculator covers the most common scenarios encountered in coursework, business analytics, and research.

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