Z 分数计算器 — 计算 Z 分数、概率 & 百分位
Z 分数衡量某值距均值的标准差数：z = (x − μ) / σ。输入您的数值、均值和标准差，即时计算 z 分数、累积概率和百分位——或将 z 分数反向换算回原始值。
Frequently Asked Questions
Z 分数（标准分）是什么？如何计算？
Z 分数表示某个数据点距离均值的标准差倍数：Z = (X - μ) / σ，其中 X 是数据点，μ 是均值，σ 是标准差。Z=0 表示等于均值；Z=1 表示高于均值 1 个标准差；Z=-2 表示低于均值 2 个标准差。
Z 分数有什么用途？
Z 分数的主要用途：①标准化不同量纲的数据以便比较（如比较不同科目的考试成绩）；②识别离群值（|Z| > 3 通常视为异常值）；③计算在正态分布中特定分数对应的百分位数；④用于假设检验和统计控制图（6σ 质量管理）。
如何将 Z 分数转换为百分位数？
在标准正态分布中，百分位数可通过查阅 Z 表（标准正态分布表）或使用累积分布函数（CDF）得到。例如：Z=0 → 50 百分位；Z=1 → 84.1 百分位；Z=-1 → 15.9 百分位；Z=2 → 97.7 百分位。本计算器自动显示对应百分位数。
Z 分数和 T 分数有什么区别？
Z 分数基于总体参数（总体均值 μ 和总体标准差 σ）；T 分数基于样本统计量（样本均值 x̄ 和样本标准差 s），在样本量较小（n < 30）且总体标准差未知时使用。随着样本量增大，T 分布趋近于正态分布（Z 分布）。
Z-Score Formula
A z-score (also called a standard score) measures how many standard deviations a data point is from the mean of its distribution. The formula is:
Z-Score
z = (x − μ) / σ
Where x is the observed value, μ is the population mean, and σ is the standard deviation.
Example: x = 85, μ = 70, σ = 10 → z = (85 − 70) / 10 = 1.5
Value from Z-Score (Inverse)
x = μ + z × σ
Example: z = 1.5, μ = 70, σ = 10 → x = 70 + 1.5 × 10 = 85
Standard Normal Distribution
The standard normal distribution (also called the z-distribution) is a normal distribution with a mean of 0 and a standard deviation of 1. When you convert a value to a z-score, you are standardizing it onto this distribution.
The cumulative probability (also called the cumulative distribution function, or CDF) tells you what fraction of values in the distribution fall at or belowa given z-score. This calculator uses the Abramowitz & Stegun polynomial approximation to the error function (erf) for fast, accurate results.
Key property:By the empirical rule (68–95–99.7 rule):
- ~68% of data falls within z = ±1
- ~95% of data falls within z = ±1.96
- ~99.7% of data falls within z = ±3
How to Interpret Z-Scores
Positive Z-Score
A positive z-score means the value is above the mean. For example, z = 2.0 means the value is 2 standard deviations above average, placing it in roughly the 97.7th percentile.
Negative Z-Score
A negative z-score means the value is belowthe mean. For example, z = −1.0 means the value is 1 standard deviation below average, placing it in roughly the 15.9th percentile.
Z-Score of Zero
A z-score of exactly 0 means the value equals the mean exactly, placing it at the 50th percentile.
Practical Interpretation
Z-scores are widely used in standardized testing, quality control (Six Sigma), finance (measuring returns relative to benchmarks), and research (identifying outliers, computing p-values). A value with |z| > 3 is often treated as a statistical outlier.
How to Use This Calculator
- Select a mode— choose “Find Z-Score” to convert a raw value to a z-score, or “Find Value” to convert a z-score back to the original scale.
- Enter your inputs— fill in the value (or z-score), the mean, and the standard deviation.
- Read the result instantly— the calculator shows the z-score (or value), cumulative probability, and percentile.
- Copy the results— click the Copy button to copy all results to your clipboard.
Common Z-Values Table
These critical z-values are widely used in hypothesis testing and confidence intervals:
| Z-Score | Confidence Level | Cumulative Probability | Percentile |
|---|---|---|---|
| 1.282 | 80% | 0.8997 | ~90th |
| 1.645 | 90% | 0.9500 | ~95th |
| 1.960 | 95% | 0.9750 | ~97.5th |
| 2.326 | 98% | 0.9900 | ~99th |
| 2.576 | 99% | 0.9950 | ~99.5th |
| 3.000 | 99.7% | 0.9987 | ~99.9th |
Related Tools
- Standard Deviation Calculator — Compute mean, variance, and standard deviation from a data set
- Five Number Summary Calculator — Get min, Q1, median, Q3, and max of a data set
- Percentage Calculator — Find percentages, percent change, and percent of a number
- Scientific Calculator — Full-featured scientific calculator for advanced math