📊 統計學工具

🔬 F 方差檢驗計算器F Test Variance F 方差檢驗計算器：輸入兩組樣本方差與自由度，計算 F 統計量與臨界值，檢驗兩總體方差是否相等，用於方差齊性判定。F Test Variance - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📐 V 計算器Cramers V Cramér&#x27;s V 計算器Cramers V - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📏 Z 分數計算器Z Fraction Calculator 輸入原始分數、總體均值與總體標準差，將分數標準化為 Z 分數 (x−μ)/σ，並給出對應的百分位排名，Z>0 表示高於均值、Z<0 表示低於均值，便於橫向比較不同分佈的數據。Z Fraction Calculator - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📏 d 效應量計算器Cohens D Cohen&#x27;s d 效應量計算器Cohens D - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 🧪 t 檢驗統計量T 輸入樣本均值、假設總體均值、樣本標準差與樣本量，計算單樣本 t 統計量 t=(x̄−μ)/(s/√n)，並給出自由度，|t| 越大表示樣本均值偏離假設值的證據越強，用於均值假設檢驗的第一步。T - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 二項分佈概率計算器Binomial Pmf 輸入試驗次數 n 與成功次數 k、成功概率 p，計算 n 次獨立伯努利試驗恰好 k 次成功的概率，用於離散分佈與風險建模。Binomial Pmf - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 🎰 假設檢驗 p 值近似P 輸入 Z 統計量，用正態分佈近似快速估算雙尾 Z 檢驗的 p 值，並提示 p<0.05 時通常在 5% 水平下拒絕原假設，適用於大樣本假設檢驗時快速判斷顯著性。P - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 🏔️ 偏度與峯度Statistics 7 輸入一組數據，計算偏度Skewness與峯度Kurtosis，判斷分佈是否左偏或右偏、是否比正態分佈更尖或平緩，輔助識別數據分佈形態與異常尾部特徵。Statistics 7 - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 幾何平均數計算器Geometric Mean 幾何平均數計算器：輸入數值集合，按乘積的 n 次根計算幾何平均數，適合增長率、比率等相乘型數據的集中趨勢度量。Geometric Mean - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 🏋️ 加權平均數Statistics 11 加權平均數計算器：輸入數值與對應權重，按權重自動歸一化計算加權平均數，用於成績、指標等加權彙總。Statistics 11 - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 單因素 ANOVA 計算器One Way Anova 單因素方差分析（One-way ANOVA）在線計算器，輸入多組樣本自動計算組間/組內平方和、F 值與 p 值，判斷均值差異顯著性，純前端運行。One Way Anova - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📏 單樣本 Z 檢驗計算器One Sample Z Test 輸入樣本均值、樣本量與已知的總體標準差，檢驗樣本均值是否偏離給定的假設值，輸出 Z 統計量及雙側與單側 p 值，適用於總體標準差已知時的均值顯著性檢驗。One Sample Z Test - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📐 單樣本 t 檢驗計算器One Sample T Test 輸入樣本數據（或樣本均值、標準差與樣本量），在總體標準差未知時，用樣本 t 分佈檢驗樣本均值與假設總體均值是否存在顯著差異，輸出 t 統計量、自由度和 p 值，適用於小樣本均值假設檢驗。One Sample T Test - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 卡方檢驗計算器Chi Square Test 卡方檢驗計算器：輸入觀測值與期望值頻數，計算 χ² 統計量並判斷分佈偏離顯著性，用於擬合優度與獨立性檢驗。Chi Square Test - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 變異係數比較Statistics 10 變異係數比較工具：輸入兩組數據的均值與標準差，計算變異係數 CV 消除量綱後比較離散程度，用於不同單位數據的波動性對比。Statistics 10 - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📦 四分位距 IQRIqr 輸入數據集，計算 Q1、Q2、Q3 與四分位距 IQR，並據 1.5×IQR 規則標出異常值邊界，用於箱線圖與離羣點檢測。Iqr - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📈 均值中位數眾數Statistics 12 輸入最多 7 個數值，快速計算這組數據的算術均值、中位數、眾數與極差，並給出數據總和，適合在課堂上或日常中快速描述一組數據的集中趨勢與離散程度。Statistics 12 - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 均值標準誤計算器Standard Error 均值標準誤計算器：輸入樣本標準差與樣本量，求樣本均值分佈的標準誤 SE = s/√n，用於置信區間與顯著性檢驗。Standard Error - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📐 均值樣本量計算器Sample Size Mean 均值樣本量計算器：給定置信水平、容許誤差與總體標準差，估計總體均值所需的最小樣本量，用於實驗設計。Sample Size Mean - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 均值置信區間計算器Confidence Interval 均值置信區間計算器：已知總體標準差時輸入樣本均值與樣本量，求給定置信水平下均值的置信區間，用於統計推斷。Confidence Interval - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 總體方差計算器Population Variance 總體方差在線計算器，輸入完整數據集自動求均值與離差平方平均，衡量數據離散程度，支持批量粘貼、結果實時顯示，數據本地處理。Population Variance - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 最小二乘迴歸計算器Linear Regression y = a + b·x（最小二乘）Linear Regression - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 標準分 Z 值計算器Z Score Calc 標準分 Z 值計算器：輸入原始值、均值與標準差，求標準分 Z = (x−μ)/σ，用於數據標準化與離羣判斷。Z Score Calc - calculate online, free and accurate. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics to... 📉 標準差與方差Standard Deviation and Variance 輸入一組數值並選擇類型（1 為樣本、2 為總體），計算樣本或總體標準差、方差以及變異係數 CV，用於衡量數據的離散程度與相對波動水平，便於不同量綱數據的比較。Standard Deviation and Variance - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics to... 📊 樣本方差計算器Sample Variance - : s ² = Σ( x ᵢ− x ̄)²/( n − 1 ) 樣本方差計算器：輸入數據集，按 n−1 自由度計算樣本方差，衡量數據離散程度，用於推斷統計與假設檢驗。Sample Variance - : s ² = Σ( x ᵢ− x ̄)²/( n − 1 ) - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of... 🧮 樣本量估算Sample Size Estimate 輸入可接受的誤差幅度、置信水平與總體標準差，估算估計總體均值所需的最小樣本量 n，幫助在調查或實驗中以足夠精度規劃抽樣數量，避免樣本不足導致誤差過大。Sample Size Estimate - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 正態分佈 CDF 計算器Normal Cdf 正態分佈 CDF 計算器：輸入均值、標準差與取值 x，計算標準正態分佈累積概率 P(X ≤ x)，用於概率與分位評估。Normal Cdf - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 🔔 正態分佈概率Statistics 輸入均值、標準差以及區間上下界，計算正態分佈下該區間的概率（即曲線下面積），下界可填很大的負數近似負無窮，用於求某取值範圍內的累積概率或分位數區間。Statistics - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 🎯 比例樣本量計算器Sample Size ( Ratio ) 比例樣本量計算器：給定置信水平、誤差界與預期比例，估計總體比例所需樣本量，用於抽樣調查設計。Sample Size ( Ratio ) - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 比例置信區間計算器Confidence Proportion 比例置信區間計算器：輸入樣本成功數與樣本量（或樣本比例），按正態近似求總體比例在給定置信水平下的置信區間，用於調查與醫學估計。Confidence Proportion - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 🔢 比值比計算器Odds Ratio 輸入 2×2 列聯表的四項頻數，計算比值比 OR 及其置信區間，衡量暴露與結局的關聯強度，用於流行病學與醫學統計分析。Odds Ratio - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 泊松分佈概率計算器Poisson Pmf 輸入單位時間（或空間）內事件的平均發生率 λ 和目標發生次數 k，計算泊松分佈的概率質量函數 P(X=k)，即恰好發生 k 次事件的概率，常用於客服來電、故障次數等稀有事件的建模。Poisson Pmf - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. ⚖️ 獨立雙樣本 t 檢驗計算器Independent T Test 獨立雙樣本 t 檢驗計算器：輸入兩組獨立樣本數據，計算 t 值與 p 值，判斷兩總體均值是否存在顯著差異，用於實驗與對比研究。Independent T Test - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 🥞 由兩組樣本量方差求合併方差Merge Variance Calculator - Merge Variance : Input n1 , s1 , n2 , s2 , Output s_p ² 輸入兩組樣本量n₁、n₂及其方差s₁²、s₂²，按加權平均求合併方差s_p²，用於兩獨立樣本t檢驗前的方差齊性處理，適用於實驗數據的統計推斷。Merge Variance Calculator - Merge Variance : Input n1 , s1 , n2 , s2 , Output s_p ² - free online tool. ↔️ 由臨界值與標準誤求誤差幅度Margin of Error 輸入臨界值 z 與標準誤 SE，按 ME = z×SE 計算置信區間的誤差幅度，用於抽樣調查、問卷與實驗結果的不確定性估計。Margin of Error - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. ➖ 由事件概率求其補事件概率Probability Complement 對立事件概率計算器：輸入事件概率 p，按 P(Aᶜ) = 1 − P(A) 求對立（補）事件概率，用於概率基本運算。Probability Complement - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 💯 由均值與標準差求變異係數Coefficient of Variation 輸入均值 μ 與標準差 σ，按 CV = σ/μ × 100% 計算變異係數，用於比較不同量綱數據的離散程度與相對波動水平。Coefficient of Variation - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 由數值序列求平均絕對偏差Mean Absolute Deviation 輸入一組數值，按 MAD = Σ|xᵢ − x̄|/n 計算平均絕對偏差，衡量數據平均離散程度，用於預測誤差與穩健性評估。Mean Absolute Deviation - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. ↔️ 由數值序列求極差Range Calculator 極差計算器：輸入數值序列，按 R = max − min 求極差，快速衡量數據離散跨度，用於描述統計初篩。Range Calculator - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📈 由數值序列求樣本偏度Skewness Sample 樣本偏度計算器：輸入數值序列，按三階中心矩公式求樣本偏度 g₁，判斷分佈左偏/右偏及對稱程度。Skewness Sample - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 由數值序列求樣本方差Sample Variance Calculator 輸入以逗號或空格分隔的一組數值，計算樣本方差 s²=Σ(xᵢ−x̄)²/(n−1)，作為標準差的平方衡量數據離散程度，常用於統計描述與後續檢驗的方差分析。Sample Variance Calculator - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 由數值序列求樣本標準差Sample Standard Deviation Calculator 輸入以逗號或空格分隔的一組數值，計算樣本標準差 s，反映數據相對均值的平均離散程度，按 n−1 作分母的無偏估計，適合評估一組觀測值的波動大小。Sample Standard Deviation Calculator - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statisti... #️⃣ 由樣本均值、假設均值與標準差求 t 值T Score 輸入樣本均值 x̄、假設均值 μ、標準差 s 與樣本量 n，求 t 統計量。T Score - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 💯 由樣本比例與樣本量求標準誤Standard Error Proportion 輸入樣本比例 p 與樣本量 n，按公式 SE=√[p(1−p)/n] 計算樣本比例的標準誤，衡量樣本比例估計總體比例的抽樣波動，用於比例置信區間與假設檢驗。Standard Error Proportion - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. ⬜ 由相關係數求決定係數Correlation R Squared 決定係數計算器：輸入相關係數 r，按 R² = r² 求決定係數，表示迴歸模型對因變量變異的解釋比例，用於迴歸擬合優度評估。Correlation R Squared - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. ∑ 由試驗數、成功概率與上限求累積概率Binomial Cdf 輸入試驗數 n、成功概率 p 與上限 x，按累積分佈求 P(X≤x)，用於質量抽檢、可靠性與二項分佈下累計概率的統計分析。Binomial Cdf - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 🔀 由賠率求對應概率Odds to Probability 賠率轉概率計算器：輸入賠率 o，按 p = o/(1+o) 換算對應概率，用於博彩、醫學風險與流行病學表達轉換。Odds to Probability - convert values online, free and instant. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the St... 📊 百分等級計算器Percentile Rank 百分等級計算器：輸入數值與數據集，計算該值低於總體中多少比例的數據（百分等級），用於成績與排名評估。Percentile Rank - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 皮爾遜相關係數計算器Correlation Coefficient 皮爾遜相關係數計算器：輸入兩變量成對數據，計算 Pearson 相關係數 r 衡量線性相關強度與方向，並給出顯著性判斷，用於相關分析。Correlation Coefficient - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📈 相對風險計算器Relative Risk ( RR ) 相對風險計算器：輸入暴露組與非暴露組的結局發生率，計算相對風險 RR 衡量暴露與結局的關聯強度，用於流行病學隊列研究。Relative Risk ( RR ) - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📏 線性迴歸擬合Statistics 16 輸入成對(x,y)樣本數據，用最小二乘法擬合直線y=ax+b，輸出斜率a、截距b、相關係數及擬合優度，並可代入任意x預測y值，用於趨勢分析與數據建模。Statistics 16 - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 🎯 置信區間估算Statistics 4 輸入樣本均值、總體或樣本標準差與樣本量，並指定置信水平（如 95%），用 Z 分佈估算總體均值的置信區間，給出上下限，適用於大樣本或已知總體標準差時的區間估計。Statistics 4 - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 📊 調和平均數計算器Harmonic Mean 調和平均數計算器：輸入數值集合，按倒數的算術平均之倒數計算調和平均數，適合速率、單價等倒數型數據的平均。Harmonic Mean - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 🧠 貝葉斯定理計算器Bayes Theorem 貝葉斯定理計算器：輸入先驗概率與似然，按貝葉斯公式更新後驗概率，用於醫學診斷與統計推斷。Bayes Theorem - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools. 🔗 配對 t 檢驗計算器Paired T Test 輸入同一組對象的配對測量值（如服藥前後），計算每對差值及其均值與標準差，用配對 t 檢驗判斷差值是否顯著不為零，輸出 t 統計量、自由度與 p 值，用於前後對照實驗的差異檢驗。Paired T Test - free online tool. Free online tool on ToolBox — 100% client-side, no data uploaded, no install. Part of the Statistics tools.