在R中模拟获取真实标准差所需样本量并复现对应可视化图表

你现有代码的核心问题是不同样本量对应的总体标准差不统一，你给rnorm传入的第三个参数逐次变小，所以你得到的曲线是总体标准差本身下降的结果，和“样本量增大时样本标准差收敛到真实总体标准差”的仿真目标不符。

正确实现思路 #

• 固定总体分布参数：比如设定正态分布均值为0、真实标准差为固定值，若使用t分布可先计算对应自由度下的真实标准差，按需调整缩放系数。
• 设定测试用的样本量序列，覆盖你需要观测的收敛区间。
• 每个样本量可重复抽样多次取平均，消除随机波动，让收敛曲线更平滑；如果需要展示波动范围，可保留多次抽样的分位数结果。
• 绘图时添加真实标准差的参考线，方便对比收敛效果。

代码实现 #

基础版：正态分布收敛曲线

set.seed(123)
# 仿真参数设置
true_sd <- 10 # 设定真实总体标准差
sample_sizes <- seq(1, 200, by = 1) # 测试的样本量范围
n_rep <- 100 # 每个样本量重复抽样次数，数值越大曲线越平滑

# 计算每个样本量对应的平均样本标准差
sd_result <- sapply(sample_sizes, function(n) {
  replicate(n_rep, sd(rnorm(n, mean = 0, sd = true_sd))) |> mean()
})

# 绘制结果
plot(sample_sizes, sd_result, type = "l", lwd = 2, col = "darkblue",
     xlab = "样本量", ylab = "样本标准差",
     main = "正态分布下样本标准差随样本量收敛情况")
# 添加真实标准差参考线
abline(h = true_sd, col = "red", lwd = 2, lty = 2)
legend("bottomright", legend = c("样本标准差均值", "真实总体标准差"),
       col = c("darkblue", "red"), lty = c(1,2), lwd = 2)

t分布版本

仅需要修改抽样部分的代码，调整为t分布并做缩放匹配目标真实标准差即可：

set.seed(123)
df <- 5 # t分布自由度
target_true_sd <- 10 # 目标真实标准差
# t分布原始标准差为sqrt(df/(df-2))，计算缩放系数匹配目标标准差
scale_factor <- target_true_sd / sqrt(df/(df-2))
sample_sizes <- seq(1, 200, by = 1)
n_rep <- 100

sd_result_t <- sapply(sample_sizes, function(n) {
  replicate(n_rep, sd(scale_factor * rt(n, df = df))) |> mean()
})

plot(sample_sizes, sd_result_t, type = "l", lwd = 2, col = "darkgreen",
     xlab = "样本量", ylab = "样本标准差",
     main = "t分布下样本标准差随样本量收敛情况")
abline(h = target_true_sd, col = "red", lwd = 2, lty = 2)
legend("bottomright", legend = c("样本标准差均值", "真实总体标准差"),
       col = c("darkgreen", "red"), lty = c(1,2), lwd = 2)

带波动区间的版本

如果需要复现原图中可能存在的波动范围，可绘制多次抽样的95%分位数区间：

set.seed(123)
true_sd <- 10
sample_sizes <- seq(1, 200, by = 1)
n_rep <- 100

# 保留所有重复抽样的结果
sd_all <- sapply(sample_sizes, function(n) {
  replicate(n_rep, sd(rnorm(n, mean = 0, sd = true_sd)))
})
# 计算均值和95%分位区间
sd_mean <- apply(sd_all, 2, mean)
sd_q025 <- apply(sd_all, 2, quantile, 0.025)
sd_q975 <- apply(sd_all, 2, quantile, 0.975)

# 绘图
plot(sample_sizes, sd_mean, type = "l", lwd = 2, col = "darkblue",
     ylim = range(c(sd_q025, sd_q975)),
     xlab = "样本量", ylab = "样本标准差",
     main = "样本标准差收敛（含95%波动区间）")
# 绘制区间填充
polygon(c(sample_sizes, rev(sample_sizes)), c(sd_q025, rev(sd_q975)),
        col = adjustcolor("lightblue", 0.3), border = NA)
abline(h = true_sd, col = "red", lwd = 2, lty = 2)
legend("bottomright", legend = c("样本标准差均值", "95%波动区间", "真实总体标准差"),
       col = c("darkblue", "lightblue", "red"), lty = c(1,NA,2), lwd = c(2,NA,2),
       pch = c(NA,15,NA))

内容的提问来源于stack exchange，提问作者Homer Jay Simpson