🔔 中心极限定理计算器（样本均值的概率）Central Limit Theorem Calculator
输入总体均值 μ、标准差 σ 和样本量 n，按中心极限定理求样本均值 X̄ 的分布（均值 μ、标准误 σ/√n），并计算 X̄ 小于、大于或落在某个区间的概率。Enter the population mean μ, standard deviation σ and sample size n to get the sampling distribution of the mean X̄ by the central limit theorem (mean μ, standard error σ/√n), and the probability that X̄ is below, above or between given values.
使用说明与常见问题How-to & FAQ
「中心极限定理计算器（样本均值的概率）」是什么？What is the Central Limit Theorem Calculator tool?
输入总体均值 μ、标准差 σ 和样本量 n，按中心极限定理求样本均值 X̄ 的分布（均值 μ、标准误 σ/√n），并计算 X̄ 小于、大于或落在某个区间的概率。Enter the population mean μ, standard deviation σ and sample size n to get the sampling distribution of the mean X̄ by the central limit theorem (mean μ, standard error σ/√n), and the probability that X̄ is below, above or between given values.
如何使用「中心极限定理计算器（样本均值的概率）」？How do I use the Central Limit Theorem Calculator tool?
在「中心极限定理计算器（样本均值的概率）」中，填写「总体均值 μ」、「总体标准差 σ」、「样本量 n」、「求概率」、「a」等参数，结果会随输入实时计算并显示在下方。In the Central Limit Theorem Calculator: Fill in "Population mean μ", "Population standard deviation σ", "Sample size n", "Probability", "a"; results are recalculated instantly as you type.
中心极限定理说了什么？What does the central limit theorem say?
从均值为 μ、标准差为 σ 的总体中独立抽取 n 个样本，当 n 足够大时，样本均值 X̄ 近似服从正态分布，均值为 μ，标准差（标准误）为 σ/√n，不管总体本身是什么形状。Draw n independent observations from a population with mean μ and standard deviation σ: when n is large enough, the sample mean X̄ is approximately normal with mean μ and standard deviation (standard error) σ/√n, whatever the shape of the population.
样本量多大才够？How large must the sample be?
常用的经验是 n ≥ 30；总体越偏斜需要的样本越大。如果总体本身就是正态分布，任何 n 下结果都是精确的。n < 30 时工具会给出提醒。The usual rule of thumb is n ≥ 30, and more for strongly skewed populations. If the population itself is normal the result is exact for any n. A warning appears when n is below 30.
概率是怎么算的？How are the probabilities computed?
先把 a、b 换算成 z = (a − μ)/(σ/√n)，再用标准正态分布函数 Φ(z) 求概率，Φ 用 Marsaglia 的级数计算，误差小于 10⁻¹⁵；我们用 mpmath 核对过，比如 μ=100、σ=15、n=36 时 P(X̄<103) = Φ(1.2) = 0.884930。Each bound is turned into z = (a − μ)/(σ/√n) and the probability comes from the standard normal Φ(z), computed with Marsaglia's series to within 10⁻¹⁵. Checked against mpmath: with μ = 100, σ = 15, n = 36, P(X̄ < 103) = Φ(1.2) = 0.884930.
这里的 σ 是样本标准差吗？Is σ here the sample standard deviation?
应该填总体标准差。若只有样本标准差 s 且样本较小，严格来说应改用 t 分布；n 较大时用 s 代替 σ 的结果非常接近。It should be the population standard deviation. If you only have a sample standard deviation s and a small sample, the t distribution is the strict choice; for large n using s in place of σ gives nearly the same answer.
计算精度如何？How precise are the results?
「中心极限定理计算器（样本均值的概率）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Central Limit Theorem Calculator uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「中心极限定理计算器（样本均值的概率）」免费吗？需要注册吗？Is the Central Limit Theorem Calculator free? Do I need an account?
「中心极限定理计算器（样本均值的概率）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Central Limit Theorem Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.