Interactive: the Central Limit Theorem in motion - Introduction to Statistical Concepts and Methods (MATH 2200)

Drag the sample-size slider below. Each setting shows the distribution of 5,000 sample means drawn from the same right-skewed simulated population. As the sample size $n$ grows, the sampling distribution (a) stays centered on the population mean (orange dashed line) and (b) narrows -- its spread is the standard error $\text{SE} = \sigma/\sqrt{n}$ , shown in the title. Even though the population is skewed, the sample mean becomes bell-shaped. That is the Central Limit Theorem.

import numpy as np
import plotly.graph_objects as go
from IPython.display import HTML

# Okabe-Ito colorblind-safe palette
OI_BLUE   = "#0072B2"   # histogram bars (sample means)
OI_ORANGE = "#D55E00"   # population-mean reference line

# A skewed, strictly-positive "population" standing in for daily PM2.5-style
# readings (gamma is right-skewed, like real air-quality data). SIMULATED data,
# fixed seed -> identical every build. No real dataset value is claimed here.
rng = np.random.default_rng(2200)
pop = rng.gamma(shape=1.5, scale=6.2, size=200_000)
mu, sigma = pop.mean(), pop.std()

sizes = [2, 5, 10, 20, 30, 50, 100]   # the sample-size slider stops
reps  = 5000                          # repeated samples per slider stop

# One histogram trace per sample size; only the first is visible at load.
fig = go.Figure()
for k, n in enumerate(sizes):
    means = rng.choice(pop, size=(reps, n)).mean(axis=1)
    fig.add_trace(go.Histogram(
        x=means, visible=(k == 0), nbinsx=45,
        name=f"n={n}", marker_color=OI_BLUE, opacity=0.9,
        hovertemplate="sample mean: %{x:.2f}<br>count: %{y}<extra></extra>",
    ))

# Population mean as a layout line (always visible, not a toggled trace).
fig.add_vline(x=mu, line_dash="dash", line_color=OI_ORANGE, line_width=2,
              annotation_text="population mean", annotation_position="top right")

# Slider: show exactly one sample-size's histogram and update the title/SE.
steps = []
for k, n in enumerate(sizes):
    se = sigma / np.sqrt(n)
    steps.append(dict(
        method="update", label=str(n),
        args=[{"visible": [c == k for c in range(len(sizes))]},
              {"title.text": f"Sampling distribution of the sample mean "
                             f"(simulated)  -  n = {n},  SE = sigma/sqrt(n) "
                             f"= {se:.2f}"}]))

fig.update_layout(
    template="simple_white", bargap=0.02, showlegend=False,
    xaxis_title="Sample mean of the simulated population",
    yaxis_title="Number of samples (out of 5000)",
    xaxis_range=[0, 20],   # FIXED range so bars don't jump as n changes
    height=460, margin=dict(t=70, r=20, b=60, l=70),
    sliders=[dict(active=0, pad={"t": 40},
                  currentvalue={"prefix": "Sample size  n = "}, steps=steps)],
    title=f"Sampling distribution of the sample mean (simulated)  -  n = {sizes[0]},  "
          f"SE = sigma/sqrt(n) = {sigma/np.sqrt(sizes[0]):.2f}",
)

# THE GOTCHA FIX: emit text/html ourselves (embeddable div + CDN plotly.js).
HTML(fig.to_html(full_html=False, include_plotlyjs="cdn"))