← ToolBox / 假设检验 p 值近似

🔬 P-Value from z-Score Estimator

Enter a z statistic to approximate the two-sided p value using the normal distribution, with a note that p below 0.05 usually rejects the null hypothesis at the 5% level.

Z 统计量

📐 计算公式与说明

Z 统计量 p 值（本页代码口径，erf 数值实现正态分布 Φ）：单尾 p=1−0.5×(1+erf(|z|÷√2))；双尾 p=2×单尾，两项均保留 6 位小数。校验：无强校验，空值按 0 兜底。判据：无固定分级，|z| 越大 p 越小，通行口径 p<0.05 视为统计显著。

p < 0.05 通常认为在 5% 水平显著。

Formula & Standard References

Item	Formula / Note	Source / Standard
Two-tailed p-value for a Z test	p = 2 · (1 - Phi(\|z\|)) = 2 · Phi(-\|z\|), where Phi is the standard normal CDF	NIST/SEMATECH e-Handbook of Statistical Methods §1.3.5.2
Significance convention	Reject the null hypothesis when p < alpha, commonly alpha = 0.05	Fisher RA, Statistical Methods for Research Workers (1925)
Interpretation of a p-value	The probability, under H0, of a result at least as extreme as observed	Wasserstein RL, Lazar NA (2016), The ASA statement on p-values, The American Statistician
Normal approximation premise	Requires known sigma or a large sample so the mean is approximately normal	NIST/SEMATECH e-Handbook of Statistical Methods §1.3.5.2

📚 深度解析：双尾 p 值（标准正态）

💡 常见使用场景

给定 z 求双尾 p=2(1−Φ(|z|))。
假设检验显著性判定。
p<0.05 通常判显著。

z=1.96

p=2(1−Φ(1.96))=2×0.025=0.05，恰为 5% 显著界。

z=2.58

p=2(1−Φ(2.58))≈2×0.0049≈0.0098，约 1% 显著。

❓ 常见问题（FAQ）

p 值含义？: 在原假设成立下，观察到当前或更极端结果的概率。p 小≠效应大，只说明"不太像随机"。
为何双尾？: 无方向假设时两侧极端都算异常，双尾 p 是单尾两倍，更保守。

假设检验 p 值近似