The Normal Distribution, Interactively

Move the sliders, hover the charts and rotate the surface to build real intuition for the most important distribution in statistics.

The normal distribution appears everywhere: measurement error, heights, test scores and the sampling distribution of almost any average. Instead of memorising its formula, let us play with it.

TipWhat you will learn

• How \mu and \sigma shape the bell curve
• Where the 68–95–99.7 rule comes from
• How two correlated normals form a 3D surface

The formula

A random variable X \sim \mathcal{N}(\mu, \sigma^2) has density

f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \tag{1}

Explore the parameters

Drag the sliders. The shaded band is \mu \pm k\sigma, and the readout shows the probability it contains.

import {ms, msStyle} from "/assets/js/ms-plot.js"
viewof mu    = Inputs.range([-3, 3],   {value: 0,   step: 0.1,  label: "Mean μ"})
viewof sigma = Inputs.range([0.6, 2.5], {value: 1,   step: 0.05, label: "Std dev σ"})
viewof k     = Inputs.range([0.5, 3],  {value: 1,   step: 0.5,  label: "Band ±kσ"})

pdf = x => Math.exp(-((x - mu) ** 2) / (2 * sigma ** 2)) / (sigma * Math.sqrt(2 * Math.PI))
erf = x => { const t = 1 / (1 + 0.3275911 * Math.abs(x));
  const y = 1 - (((((1.061405429 * t - 1.453152027) * t) + 1.421413741) * t - 0.284496736) * t + 0.254829592) * t * Math.exp(-x * x);
  return x >= 0 ? y : -y; }
prob = erf(k / Math.SQRT2)
curve = d3.range(-7, 7.01, 0.02).map(x => ({x, y: pdf(x)}))
band  = curve.filter(d => d.x >= mu - k * sigma && d.x <= mu + k * sigma)

html`<div class="ms-readout"><span>P(μ − ${k}σ ≤ X ≤ μ + ${k}σ)</span><b>${(prob * 100).toFixed(1)}%</b></div>`

Plot.plot({
  ...msStyle,
  x: {domain: [-7, 7], label: "x"},
  y: {domain: [0, 0.7], label: "Density"},
  marks: [
    Plot.areaY(band, {x: "x", y: "y", fill: ms.indigo, fillOpacity: 0.18}),
    Plot.line(curve, {x: "x", y: "y", stroke: ms.indigo, strokeWidth: 2.5}),
    Plot.ruleX([mu], {stroke: ms.indigo, strokeDasharray: "4,4", strokeOpacity: 0.7}),
    Plot.ruleY([0], {strokeOpacity: 0.25}),
    Plot.tip(curve, Plot.pointerX({x: "x", y: "y", title: d => `x = ${d.x.toFixed(2)}\nf(x) = ${d.y.toFixed(3)}`}))
  ]
})

NoteThe 68–95–99.7 rule

Set the band to 1\sigma, 2\sigma and 3\sigma. The readout gives about 68.3%, 95.4% and 99.7%, no matter what \mu and \sigma are.

Samples versus theory

Hover over the bars to read exact values. Drag to zoom and double-click to reset.

import numpy as np
import plotly.graph_objects as go
from scipy import stats
from ms_charts import COLORS, PLOTLY_CONFIG   # also registers the "mlstation" Plotly theme

rng = np.random.default_rng(0)
samples = rng.normal(0, 1, 10_000)
x = np.linspace(-4, 4, 400)

fig = go.Figure()
fig.add_histogram(x=samples, histnorm="probability density", nbinsx=70, name="Samples",
                  marker=dict(color=COLORS["indigo"], opacity=0.35, line=dict(width=0)))
fig.add_scatter(x=x, y=stats.norm.pdf(x), mode="lines", name="Theory",
                line=dict(color=COLORS["indigo"], width=3))
fig.update_layout(height=360, xaxis_title="x", yaxis_title="Density", bargap=0.02,
                  hovermode="x unified")
fig.show(config=PLOTLY_CONFIG)

Figure 1: 10,000 draws from a standard normal, with the theoretical density overlaid.

Two variables at once

With correlation \rho, two normals form a bivariate surface. Rotate it by dragging.

f(x, y) = \frac{1}{2\pi\sqrt{1-\rho^2}} \exp\!\left(-\frac{x^2 - 2\rho xy + y^2}{2(1-\rho^2)}\right)

import numpy as np
import plotly.graph_objects as go
from ms_charts import SURFACE_SCALE, PLOTLY_CONFIG

rho = 0.6
g = np.linspace(-3, 3, 80)
X, Y = np.meshgrid(g, g)
Z = np.exp(-(X**2 - 2*rho*X*Y + Y**2) / (2*(1 - rho**2))) / (2*np.pi*np.sqrt(1 - rho**2))

fig = go.Figure(go.Surface(x=X, y=Y, z=Z, colorscale=SURFACE_SCALE, showscale=False,
                           contours=dict(z=dict(show=True, usecolormap=True, project_z=True))))
fig.update_layout(height=420, margin=dict(l=0, r=0, t=0, b=0),
                  scene=dict(xaxis_title="x", yaxis_title="y", zaxis_title="density",
                             camera=dict(eye=dict(x=1.5, y=1.3, z=0.75)), aspectratio=dict(x=1, y=1, z=0.6),
                             xaxis=dict(backgroundcolor="rgba(0,0,0,0)"),
                             yaxis=dict(backgroundcolor="rgba(0,0,0,0)"),
                             zaxis=dict(backgroundcolor="rgba(0,0,0,0)")))
fig.show(config=PLOTLY_CONFIG)

Figure 2: Bivariate normal density with correlation ρ = 0.6.

Checking normality

A Q–Q plot compares sample quantiles with theoretical ones. Points on the line mean the data is close to normal. This chart is a static image, drawn with the shared MLStation style.

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
from ms_charts import style_matplotlib, COLORS
style_matplotlib()

rng = np.random.default_rng(3)
fig, axes = plt.subplots(1, 2, figsize=(8, 3.4), sharey=False)
for ax, data, title, c in [(axes[0], rng.normal(size=300), "Normal sample", COLORS["indigo"]),
                           (axes[1], rng.exponential(size=300), "Skewed sample", COLORS["rose"])]:
    (osm, osr), (slope, icpt, _) = stats.probplot(data)
    ax.scatter(osm, osr, s=10, color=c, alpha=.7, linewidths=0)
    ax.plot(osm, slope * osm + icpt, color="#8A93A3", lw=1.2, ls="--")
    ax.set_title(title); ax.set_xlabel("Theoretical quantiles")
axes[0].set_ylabel("Sample quantiles")
plt.tight_layout(); plt.show()

Figure 3: Normal data follows the line; skewed data bends away from it.

Summary

• \mu shifts the curve, \sigma stretches it, and the area always equals 1.
• About 68%, 95% and 99.7% of values fall within 1, 2 and 3 standard deviations.
• Use Q–Q plots, not just histograms, to check normality.