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Statistics

14 calculators — Compute mean, median, standard deviation, probability, and regression

Statistics calculators for mean, median, mode, standard deviation, variance, probability distributions, regression analysis, and hypothesis testing. For students, analysts, and researchers.

Binomial Distribution

Binomial Distribution Calculator

Calculate binomial probability P(X = k) and cumulative P(X ≤ k) in seconds. Enter number of trials, probability of success, and number of successes to get results.

Chi-Square Test

Chi-Square Test Calculator

Run a chi-square goodness-of-fit test in seconds. Enter up to six rows of observed vs expected frequencies to get the chi-square statistic, degrees of freedom, and verdict.

Coefficient of Variation

Coefficient of Variation Calculator

Calculate the coefficient of variation (CV) from a mean and standard deviation instantly. Compare relative variability across datasets with different units or scales.

Confidence Interval

Confidence Interval Calculator

Calculate a confidence interval from a sample mean, standard deviation, and sample size in seconds. Get the lower bound, upper bound, and margin of error instantly.

Correlation Coefficient

Correlation Coefficient Calculator

Calculate the Pearson correlation coefficient (r) between two variables in seconds. Enter paired X, Y data points to measure the strength and direction of their relationship.

Covariance

Covariance Calculator

Calculate sample and population covariance between two variables in seconds. Enter paired X, Y data points to measure how the two variables move together instantly.

IQR

Interquartile Range (IQR) Calculator

Calculate Q1, Q3, and the interquartile range (IQR) for any dataset instantly. Enter your numbers to find the spread of the middle 50% of your data and outlier fences.

Linear Regression

Linear Regression Calculator

Calculate the simple linear regression line for paired X, Y data in seconds. Get the slope, intercept, and R² so you can predict Y from any new X value instantly.

Margin of Error

Margin of Error Calculator

Calculate the margin of error for a survey or sample estimate in seconds. Enter confidence level, standard deviation, and sample size to get the exact margin instantly.

Normal Distribution

Normal Distribution Calculator

Calculate z-score and cumulative probability for any point on a normal distribution. Enter mean, standard deviation, and x-value to get P(X ≤ x) instantly, free.

Percentile Rank

Percentile Rank Calculator

Find what percentile a value falls at within any dataset. Enter your numbers and a target value to instantly see the percentile rank and counts below or above it.

Poisson Distribution

Poisson Distribution Calculator

Calculate Poisson probability P(X = k) for any event count in seconds. Enter the average rate (lambda) and number of events to get exact and cumulative probabilities.

Sample Size

Sample Size Calculator

Find the minimum sample size needed for a survey or experiment in seconds. Enter confidence level, margin of error, and estimated standard deviation to get your answer.

T-Test

T-Test Calculator

Run a two-sample t-test in seconds. Enter each group's mean, standard deviation, and sample size to get the t-statistic, degrees of freedom, and a significance verdict.

About Statistics Calculators

Statistics calculators bring rigour and efficiency to data analysis. Whether you are a student working through a statistics assignment, a researcher analysing survey data, a business analyst testing whether a marketing campaign had a significant effect, or a quality engineer assessing production consistency, the ability to calculate statistical metrics accurately is a fundamental quantitative skill.

Descriptive statistics

Describing a dataset clearly — its centre, spread, shape, and outliers — is the essential first step in any data analysis. Mean and standard deviation are the workhorses of descriptive statistics, but understanding when they are appropriate (roughly symmetric distributions) versus when median and IQR are more informative (skewed data, outliers) is what separates rigorous analysis from misleading summaries. Our descriptive calculators compute the full set of summary statistics so you can choose the right ones for your data.

Probability and distributions

The normal distribution underpins an enormous range of statistical methods because of the Central Limit Theorem: regardless of the population distribution, the distribution of sample means approaches normality as sample size grows. Binomial and Poisson distributions model count data — defects per unit, customer arrivals per hour, exam pass/fail outcomes — and are essential in quality control, insurance, and operations research.

Hypothesis testing

Hypothesis testing is the formal framework for deciding whether observed data provides sufficient evidence against a null assumption. The choice of test — z-test vs t-test, one-sample vs two-sample, parametric vs non-parametric — depends on your data structure, sample size, and the distributional assumptions you can reasonably make. Understanding p-values correctly (the probability of seeing your data or more extreme data if the null is true) is critical; a common misinterpretation is treating a p-value as the probability that the null hypothesis is false.

Regression and correlation

Regression analysis quantifies relationships between variables and allows prediction. From simple linear regression to multiple regression and logistic regression, these models are foundational in research, machine learning, and business analytics. Correctly interpreting regression output — understanding what R², coefficients, and residuals tell you — separates meaningful analysis from spurious pattern-matching in data.

Frequently Asked Questions

What statistics calculators are available on thecalcu.com?

thecalcu.com's statistics category includes calculators for mean, median, mode, standard deviation, variance, probability distributions (normal, binomial, Poisson), hypothesis testing (z-test, t-test, chi-square), confidence intervals, correlation, linear regression, and sample size estimation. These tools are built for students, data analysts, researchers, and anyone working with data.

What is the difference between mean, median, and mode?

Mean is the arithmetic average — sum of all values divided by count. Median is the middle value when data is sorted in order; for even-count datasets, it is the average of the two middle values. Mode is the most frequently occurring value. For a dataset like [10, 15, 15, 20, 100]: mean = 32, median = 15, mode = 15. The mean is heavily influenced by the outlier (100); the median and mode are more robust. In skewed distributions (like income data), median is usually the more meaningful measure of central tendency.

How do I calculate standard deviation and variance?

Variance = Σ(xᵢ − x̄)² ÷ (n − 1) for a sample; ÷ n for a population. Standard deviation (SD) = √Variance. For data [4, 7, 13, 2]: mean = 6.5; deviations = [−2.5, 0.5, 6.5, −4.5]; squared deviations = [6.25, 0.25, 42.25, 20.25]; sum = 69; sample variance = 69 ÷ 3 = 23; sample SD = √23 ≈ 4.80. SD tells you the typical spread around the mean in the original units — a low SD means data points cluster tightly around the mean; a high SD means they are widely spread.

What is a normal distribution and how do I use z-scores?

A normal distribution is a symmetric, bell-shaped distribution characterised by its mean (μ) and standard deviation (σ). The 68-95-99.7 rule: 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ. A z-score converts any value to its distance from the mean in standard deviation units: z = (x − μ) ÷ σ. A z-score of 1.96 corresponds to the 97.5th percentile, which is why 95% confidence intervals use ±1.96σ. Z-scores allow comparison of values from different distributions.

How do I calculate a confidence interval?

95% Confidence Interval = x̄ ± (z* × SE), where x̄ is the sample mean, z* = 1.96 for 95% confidence, and SE = σ ÷ √n (standard error). For a sample of 100 with mean 75 and SD 10: SE = 10 ÷ √100 = 1.0; CI = 75 ± 1.96 × 1.0 = [73.04, 76.96]. Interpretation: if you repeated this sampling 100 times, approximately 95 of the resulting intervals would contain the true population mean. Wider intervals indicate more uncertainty; increasing sample size narrows the interval.

What is the difference between correlation and causation?

Correlation measures the strength and direction of a linear relationship between two variables, expressed as Pearson's r (range: −1 to +1). r = 1 is a perfect positive relationship; r = −1 is a perfect negative relationship; r = 0 means no linear relationship. Causation means one variable directly causes changes in another — which correlation alone cannot establish. Ice cream sales and drowning rates are highly correlated, but the cause of both is summer heat. Establishing causation requires controlled experiments, not just correlation analysis.

How do I perform a t-test?

A one-sample t-test tests whether a sample mean differs significantly from a known value: t = (x̄ − μ₀) ÷ (s ÷ √n). A two-sample t-test compares means of two independent groups: t = (x̄₁ − x̄₂) ÷ √(s₁²/n₁ + s₂²/n₂). Compare the calculated t-statistic to the critical value from the t-distribution for your degrees of freedom and significance level (usually α = 0.05). If |t| > critical value, the difference is statistically significant. A paired t-test is used when the same subjects are measured twice (before-after studies).

How is the chi-square test used?

The chi-square (χ²) test compares observed frequencies to expected frequencies: χ² = Σ[(O − E)² ÷ E], where O is observed and E is expected frequency. It is used for: (1) goodness-of-fit tests (does data fit a theoretical distribution?); (2) tests of independence (are two categorical variables related?). For a 2×2 contingency table, degrees of freedom = (rows − 1) × (columns − 1) = 1. If χ² > critical value from the table (e.g., 3.841 for df=1 at α=0.05), the variables are not independent.

How do I calculate sample size for a study?

For estimating a population mean: n = (z* × σ ÷ E)², where z* is the z-score for your confidence level (1.96 for 95%), σ is the population standard deviation, and E is the acceptable margin of error. For estimating a proportion: n = z*² × p(1−p) ÷ E². If the true proportion is unknown, use p = 0.5 to get the most conservative (largest) sample size. For a 95% CI with a margin of error of 5% and p = 0.5: n = 1.96² × 0.25 ÷ 0.0025 = 384.16, so n = 385. Larger samples give narrower confidence intervals and more statistical power.

What is the difference between Type I and Type II errors in hypothesis testing?

A Type I error (false positive) occurs when you reject a true null hypothesis — concluding there is an effect when there isn't one. Its probability is the significance level α (typically 0.05 or 5%). A Type II error (false negative) occurs when you fail to reject a false null hypothesis — missing a real effect. Its probability is β; statistical power = 1 − β (typically targeted at 80% or 90%). There is a trade-off: reducing α (being more strict) increases the risk of Type II errors. The right balance depends on the consequences of each type of error.

How do I interpret linear regression output?

Simple linear regression fits the equation ŷ = a + bx, where b is the slope (change in y per unit change in x) and a is the intercept (predicted y when x = 0). R² (coefficient of determination) measures how much of the variance in y is explained by x — R² = 0.80 means the model explains 80% of the variation. The p-value for each coefficient tests whether it is significantly different from zero. A low p-value (< 0.05) for the slope confirms a statistically significant linear relationship. Always check residual plots — patterns in residuals indicate that the linear model may not be appropriate.