統計

使用線性內插法求任何資料集的百分位數值。免費的線上百分位數計算器，將你的數字排序並回傳任一選定百分位處的值。

立即計算任何資料集的母體與樣本標準差、變異數與平均數。免費的線上標準差計算器，附逐步結果。

從平均數與標準差計算某值的 z 分數（標準分數）。免費的線上 z 分數計算器，顯示一點距離平均數有幾個標準差。

Calculate classification accuracy, precision, recall, F1 score, specificity and NPV from a confusion matrix (TP, TN, FP, FN).

Run a one-way ANOVA F-test from group summary statistics (n, mean, SD) and get the F statistic, p-value, and effect size.

使用靈敏度和偽陽性率，透過貝葉斯法則將先驗機率更新為後驗機率。

計算 n 次試驗中恰好、至多或至少 k 次成功的二項機率，以及平均值和變異數。

計算群體中至少兩人同天生日的機率，適用於任意群體規模和天數。

從牌組中不放回地抽取 n 張牌，計算恰好抽到 k 張或至少抽到一張有利牌的機率。

使用卡方檢定測試觀測次數是否與預期次數或比例相符。

從列聯表檢定兩個類別變數之間的關聯性。

從資料或摘要統計計算變異係數（CV = SD/mean）的百分比。

計算 n 次投擲（公平或有偏）中恰好、至少或至多 k 次正面朝上的機率。

從聯合機率和 P(B) 計算 P(A|B)，並檢查兩個事件是否獨立。

使用 z 分布計算平均數的信賴區間。免費的線上信賴區間計算器，適用於 90%、95% 與 99% 水準，顯示誤差界限、下界與上界。

計算兩組資料之間的皮爾森相關係數（r），以衡量其線性關係的強度與方向。免費的線上相關係數計算器，附 r² 與斜率。

計算兩個等長變數 x 和 y 之間的樣本或母體共變異數。

Calculate all nine deciles (D1–D9) for any data set. Deciles divide data into ten equal parts at the 10th, 20th, … 90th percentiles.

計算任何資料集的平均數、中位數、眾數、全距、變異數與標準差。

Calculate the expected value, min, max, variance, and standard deviation for any dice expression (NdS + modifier).

計算任意數量和類型的骰子擲出目標點數總和的機率，包括恰好、至少或至多的情況。

將 68-95-99.7 法則應用於常態資料集：獲取一、二和三個標準差的區間及百分比。

從離散機率分佈表中計算期望值、變異數和標準差。

計算給定速率的指數機率 P(X<x) 和 P(X>x)，以及平均值和變異數。

從資料集獲取最小值、Q1、中位數、Q3、最大值和四分位距，用於箱形圖摘要。

計算首次成功發生於第 k 次試驗的機率，以及累積機率、均值與變異數。

從大小為 N 的有限母體中不放回抽取 n 個，抽到 k 個成功的機率。

計算四分位距、Tukey 圍欄，並標記資料集中的離群值。

Find the interquartile range, quartiles, and five-number summary for any data set — with a normal-distribution visualization of the IQR.

以最小平方法配適直線：斜率、截距、R 平方、方程式及預測值。

Calculate lottery jackpot odds and prize-tier probabilities for any pool size and draw count using combinations.

以選定的信賴水準計算均值或比例的誤差範圍。

Calculate the mean absolute deviation, mean, median, range, and standard deviation of any data set.

Find the median, Q1, Q3, interquartile range, mode and range of any dataset — enter numbers separated by commas.

Calculate the midrange of a data set — the average of the minimum and maximum values — instantly from any list of numbers.

Find the mode (most frequently occurring value) of any dataset, with frequency breakdown and support for bimodal and multimodal data.

Calculate Mean Squared Error (MSE), Root MSE (RMSE), Mean Absolute Error (MAE), and R² from actual and predicted values.

第 r 次成功之前發生 k 次失敗的機率，含累積選項、均值與變異數。

將 z 分數轉換為面積及面積轉換為 z，並求任何常態曲線的 P(X<x)、P(X>x) 或區間。

Convert between probability, decimal odds, fractional odds, and American/moneyline odds in one step.

將賠率轉換為機率，以及機率轉換為賠率（正反方向），並提供清晰的視覺分解。

使用 z 檢定測試樣本比例是否與假設比例不同。

使用單樣本 t 檢定測試樣本均值是否與假設值不同。

使用資料列表的單因子 ANOVA F 檢定比較三個或更多組均值。

Detect outliers in a data set using the Z-score method — flag any value more than k standard deviations from the mean.

將 z、t、卡方或 F 檢定統計量轉換為任何尾部的 p 值。

使用配對 t 檢定測試配對/前後測量的均值差異。

為任何 n 與 r 計算排列（nPr）與組合（nCr）。免費的線上排列組合計算器，使用階乘並立即出結果。

在給定平均速率 lambda 的情況下計算泊松機率 P(X=k)、至多或至少 k 次事件。

計算兩個獨立事件的機率：P(A 且 B)、P(A 或 B) 與餘事件 P(非 A)。免費的線上機率計算器，適用於單一與組合事件。

Find Q1, Q2 (median), Q3, IQR and Tukey outlier fences for any data set — with a normal-distribution visualization and five-number summary.

Calculate the statistical range of a data set — the difference between the maximum and minimum values — plus mid-range and mean.

Calculate residuals (observed − predicted) for a set of data points. Get RSS, MSE, RMSE and a residual plot to diagnose model fit.

Calculate Relative Risk (RR), Odds Ratio (OR), Absolute Risk Difference and Number Needed to Treat/Harm from a 2×2 contingency table.

Calculate Relative Standard Error (RSE = SE / Mean × 100%) to assess the reliability of a survey or sample estimate.

找出在目標誤差範圍內估計母體均值所需的樣本大小。

計算在選定誤差範圍內估計比例所需的樣本大小。

Calculate Simpson's Diversity Index (D) plus Shannon entropy and Pielou's evenness from species or category counts.

從資料清單或樣本標準差與樣本大小計算平均數的標準誤差（SEM）。

使用合併雙比例 z 檢定比較兩個獨立樣本比例。

使用 Welch 或合併雙樣本 t 檢定比較兩個獨立組別的均數。

[a,b] 上的連續均勻機率：立即求得 P(c<X<d)、均數和變異數。

計算兩個事件的補集、聯集和交集機率，適用於獨立事件或已知聯合機率的情況。

從值與權重的配對計算加權平均（加權平均數）。免費的線上加權平均計算器，適用於成績、GPA、價格與調查資料。

Calculate the mean x̄ = Σ m·p(x), variance, and standard deviation from a grouped frequency distribution using class midpoints.

Calculate the 5-number summary (min, Q1, median, Q3, max), IQR, range, and Tukey outlier fences from any data set.

Calculate A/B test statistical significance: enter visitor and conversion counts for control and variant to get the p-value, z-statistic, and relative lift.

Propagate absolute uncertainties through addition, subtraction, multiplication, and division using in-quadrature rules.

Sort any list of numbers in ascending or descending order instantly — also shows min, max, range and median.

Check whether a dataset conforms to Benford's Law using a chi-square test and MAD — useful for fraud detection and data validation.

Explore the Bertrand box paradox: set box counts (GG, GS, SS) and calculate the conditional probability that the second coin is gold given the first was gold.

Explore the Bertrand paradox: three valid definitions of a random chord in a circle yield three different probabilities.

Compute the PDF, CDF, mean, variance and mode of the Beta distribution for any shape parameters α and β.

Find the minimum group size for a target probability of two people sharing a birthday — the counterintuitive birthday paradox solved with interactive math.

Calculate the Bonferroni-corrected p-value threshold for multiple hypothesis tests to control the family-wise error rate (FWER).

Calculate min, Q1, median, Q3, max, IQR and outlier fences from a dataset, with a box-plot number line.

Explore the boy-or-girl probability paradox: how knowing a child is a boy changes the probability that both children are boys.

Calculate the standard error of the mean and sampling distribution using the Central Limit Theorem. Shows 68%, 95%, and 99.7% intervals.

Apply Chebyshev's inequality to find the minimum proportion of data within k standard deviations of the mean — valid for any probability distribution.

Calculate the class width for a grouped frequency distribution: (maximum − minimum) ÷ number of classes, with rounding guidance.

Calculate R² from actual and predicted values. Shows SS_res, SS_tot, residual stats and an actual-vs-predicted plot.

Calculate Cohen's d, Hedges' g and Glass's Δ from two group means, SDs and sample sizes.

Find the probability of getting at least one streak of k consecutive heads in n coin flips (fair or biased).

Simulate flipping a fair or biased coin up to 200 times and see the heads/tails breakdown, longest streaks and distribution.

Calculate Matthews Correlation Coefficient (MCC), Cohen's Kappa and a full confusion matrix from TP, TN, FP and FN counts.

Find k in the direct proportion y = kx from any (x, y) pair and predict y for any new x value.

Apply continuity correction to approximate a binomial probability using the normal distribution. Enter n, p and k to get the corrected z-scores and probability.

Compute Pearson r, r², regression slope, and t-statistic between two data sets with a live scatter plot and regression line.

Find the critical value for z, t, chi-square, or F distributions given a significance level and tail type.

Fit a cubic polynomial y = dx³ + cx² + bx + a to your data using least-squares regression. Shows R², all 4 coefficients and a prediction.

Convert decimal (European) odds to payout, profit and implied win probability for any stake.

Calculate degrees of freedom for one-sample t-test, two-sample t-test, Welch's t-test, one-way ANOVA and chi-square tests.

Find the expected average, standard deviation, min and max for rolling any number of dice — d4 through d100 or custom sides.

Calculate all key measures of statistical dispersion — range, variance, standard deviation, IQR and MAD — from a list of numbers.

Enter a data set to compute mean, median, mode, range and standard deviation, with a distribution curve showing data spread and key markers.

Convert between effect size measures: Cohen's d, Pearson r, eta-squared, Cohen's f, odds ratio and probability of superiority.

Propagate measurement uncertainties through addition, subtraction, multiplication, division, and power operations using standard error-propagation formulas.

Predict a future value using two known data points by fitting a continuous exponential growth model P(t) = P₀·e^(rt).

Fit an exponential curve y = a·e^(bx) to data points using least-squares linearisation and compute R² goodness-of-fit.

Calculate the F-statistic and two-tailed p-value to test whether two sample variances are significantly different.

Calculate false positive rate, false discovery rate, PPV and NPV for any diagnostic test given prevalence, sensitivity and specificity.

See how a rare condition causes most positive test results to be false positives, even with a highly accurate test.

Calculate Q1 (the first quartile / 25th percentile) for any data set using the Tukey hinges method, with the five-number summary and IQR.

Compute the exact p-value for a 2×2 contingency table using Fisher's exact test, plus odds ratio and 95% CI.

Calculate the weighted average star rating, Bayesian average and positive review percentage from a 5-star distribution.

Convert fractional betting odds (e.g. 5/2) to decimal odds, implied probability, and potential payout.

Build a grouped frequency distribution from any data set: class boundaries, frequencies, relative frequencies and cumulative counts.

Plot a frequency polygon and compute the weighted mean and standard deviation from grouped data class midpoints and frequencies.

Arrange any list of numbers from greatest to least (descending order) instantly, with min, max, range, mean and median.

Build a frequency histogram from any dataset: enter comma-separated values, choose the number of bins, and see counts, relative frequencies, mean and standard deviation.

Perform a one-sample z-test: enter sample mean, population mean, std dev and n to get z-statistic, p-value and a reject/fail-to-reject decision.

Convert American, decimal or fractional betting odds to implied probability and strip out the bookmaker vig.

Calculate the Index of Qualitative Variation (IQV) to measure diversity or dispersion across nominal categories — from 0 (no variation) to 1 (maximum variation).

Find the z-score or x value for a given cumulative probability under a normal distribution. Enter p, mean and standard deviation.

Calculate the joint probability P(A∩B), the union P(A∪B), and conditional probabilities P(A|B) and P(B|A) for two events.

Sort a list of numbers from least to greatest (ascending order) and see the min, max, range, median, and mean.

Sort a list of decimal numbers from least to greatest with preserved precision. Also shows min, max, range, median, and mean.

Calculate the mean, median, mode, variance, PDF and CDF of a lognormal distribution from its log-mean μ and log-standard deviation σ.

Calculate the lower fence (Q1 − 1.5 × IQR) and upper fence to identify statistical outliers in a dataset using Tukey's box-plot method.

Compute the Mann-Whitney U statistic, z-score and two-tailed p-value for two independent groups using the normal approximation.

Run McNemar's chi-square test on a paired 2×2 table to compare two proportions from the same subjects.

Calculate mean, median, mode, range and standard deviation for any list of numbers. Paste or type your data set.

Calculate the median absolute deviation (MAD) of a dataset — a robust alternative to standard deviation that resists outliers.

Find the minimum and maximum values in any data set, plus range, mean, and median — with a number line visualization.

Convert moneyline odds to implied probability, decimal odds, fractional odds, and calculate your potential payout on any wager.

Calculate the exact win probability for staying or switching in the Monty Hall problem — generalised to any number of doors and host reveals.

Approximate binomial probabilities P(X ≤ k), P(X ≥ k), or P(X = k) using the normal distribution with continuity correction.

Calculate P(A or B) — the probability that at least one of two events occurs — for independent or mutually exclusive events.

Arrange any list of numbers — including fractions like 1/2 and 3/4 — from least to greatest, and show their positions on a number line.

Sort a list of decimal numbers from smallest to largest and see them aligned by decimal place — useful for comparing prices, measurements or test scores.

Sort any list of numbers in ascending or descending order and assign a rank to each value — tied numbers receive an average rank automatically.

Calculate p-hat (sample proportion), standard error, margin of error, and confidence interval for a proportion.

Explore Parrondo's paradox: calculate the expected gain per step of two individually losing games and their winning alternating combination.

Calculate the total number of possible password combinations, entropy in bits, and estimated brute-force crack time.

Compute the Pearson correlation coefficient r between two variables with step-by-step working, r², t-statistic, and significance test at α = 0.05.

Find the percentile rank of a score in a dataset — what percentage of values fall at or below it. Uses the (L + 0.5E) / N formula.

Find the central angle of a pie chart slice in degrees — enter the slice value and the total, and get the sector angle and percentage share.

Enter comma-separated values and labels to instantly calculate the percentage and degree angle for each slice of a pie chart.

Convert a pie chart sector angle (in degrees) to a percentage share — enter the angle and get the percentage, radians, and remainder.

Calculate the point estimate (sample mean), standard error, margin of error and confidence interval for a population mean.

Fit a polynomial curve of any degree (1–5) to your data using least-squares regression and compute R² and RMSE.

Calculate population variance (σ²), population standard deviation (σ), and sample variance (s²) from a list of numbers.

Calculate the number of possible combinations when choosing r items from n — without replacement (standard nCr) or with replacement (repetition allowed).

Convert pre-test probability to post-test probability using sensitivity, specificity and the likelihood ratio. Works for positive and negative test results.

Convert favorable and total outcomes into a probability fraction, percentage, and odds — with complement and odds-for display.

Calculate Cp and Cpk process capability indices from specification limits, process mean and standard deviation. Includes sigma level and out-of-spec estimate.

Fit a least-squares parabola y = cx² + bx + a to your data. Outputs R², all 3 coefficients, the equation and an optional prediction.

Fit a degree-4 polynomial y = ex⁴ + dx³ + cx² + bx + a to your data using least-squares regression. Shows R², all 5 coefficients and a prediction.

Convert a z-score back to the original raw score using x = μ + z·σ. Enter the z-score, mean and standard deviation to get the raw value and percentile.

Calculate the PDF, CDF, mean, mode, median and variance of the Rayleigh distribution for a given scale parameter σ and value x.

Calculate the relative error and percent error between a measured value and a true (accepted) value, plus accuracy category.

Calculate the relative frequency of an event as a proportion and percentage of total observations, plus the complement.

Calculate Relative Risk (RR), Absolute Risk Difference and a 95% confidence interval from two event-rate percentages.

Calculate roulette payouts, win probability and house edge for any bet type on American (38-pocket) or European (37-pocket) wheels.

Calculate the standard error, margin of error and confidence interval for a sample proportion using the normal approximation.

Calculate the sampling error (margin of error) for a proportion survey and find the sample size needed for ±3% or ±5% precision.

Calculate sensitivity, specificity, PPV, NPV, accuracy, F1, likelihood ratios and Youden's J from a 2×2 confusion matrix.

Calculate diagnostic sensitivity (recall / true positive rate): TP ÷ (TP + FN). Also shows the miss rate (false negative rate).

Calculate Shannon entropy H from a probability distribution — in bits, nats or hartleys — with maximum entropy and efficiency for any number of outcomes.

Compute Simpson's Diversity Index (D), Shannon entropy, Pielou's evenness, and Berger–Parker dominance from species counts.

Compute sample or population skewness of a dataset using the Fisher-Pearson coefficient, plus Pearson median skewness and excess kurtosis.

Compute Spearman's rank correlation coefficient rₛ for paired data — handles ties using average ranks.

Calculate test specificity (true negative rate) and false positive rate from true negatives and false positives.

Calculate required sample size or statistical power for a two-sample test. Uses Cohen's d, significance level α, and the z-approximation.

Build a stem-and-leaf plot from your data set and get the five-number summary instantly.

Compute the sum of squared deviations SS = Σ(xᵢ − x̄)² plus sample and population variance and standard deviation from any list of numbers.

Compute the one-sample t-statistic, degrees of freedom, standard error, and two-tailed p-value from sample mean, standard deviation, and sample size.

Calculate Q3 (the third quartile / 75th percentile) for any data set using the Tukey hinges method, with the five-number summary and IQR.

Explore the two-envelopes paradox: calculate the expected value of switching vs keeping your envelope and find the break-even probability for switching.

Calculate both Tukey fences (upper: Q3 + k × IQR, lower: Q1 − k × IQR) to find outlier thresholds for any data set.

Calculate the Upper Control Limit (UCL) and Lower Control Limit (LCL) for a Shewhart control chart using the process mean and sigma.

Calculate the upper Tukey fence (Q3 + k × IQR) from Q1 and Q3 to identify potential high outliers in a data set.

Calculate sample variance, population variance, standard deviation, coefficient of variation, and standard error from any numeric data set.

Calculate union, intersection, A-only, B-only, and conditional probabilities for two overlapping sets with a Venn diagram breakdown.

Calculate Weibull PDF, CDF, survival probability, mean, median, mode, and variance from shape k and scale λ.

Run a non-parametric Wilcoxon rank sum / Mann-Whitney U test to compare two independent groups without assuming normality.

Calculate Youden's J statistic to measure a diagnostic test's overall discriminatory ability from sensitivity and specificity.