Normality Visualization¶

Many statistical methods -- \(t\)-tests, ANOVA, linear regression -- assume that data or residuals follow a normal distribution. While formal tests like Shapiro-Wilk provide a binary decision, visual methods reveal the nature and severity of departures from normality. A histogram might show skewness, a QQ plot might show heavy tails, and these patterns inform which remedies (transformations, nonparametric alternatives) are appropriate. This page demonstrates the primary visual techniques for assessing normality.

Histogram with Normal Overlay¶

The most intuitive check overlays the fitted normal density \(f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)}\) on a normalized histogram of the data.

```python import numpy as np import matplotlib.pyplot as plt from scipy import stats

np.random.seed(42) data = np.random.normal(loc=50, scale=10, size=300)

Fit normal distribution¶

mu, sigma = stats.norm.fit(data) x = np.linspace(data.min() - 5, data.max() + 5, 200) pdf = stats.norm.pdf(x, mu, sigma)

plt.hist(data, bins=25, density=True, alpha=0.6, label='Data') plt.plot(x, pdf, 'r-', linewidth=2, label=f'Normal(\(\\mu\)={mu:.1f}, \(\\sigma\)={sigma:.1f})') plt.xlabel('Value') plt.ylabel('Density') plt.title('Histogram with Fitted Normal Density') plt.legend() plt.show() ```

Normal Probability Plot (QQ Plot)¶

The normal QQ plot plots sample quantiles against theoretical normal quantiles. If the data are normally distributed, the points follow a straight line. scipy.stats.probplot computes both the quantiles and a fitted line.

python fig, ax = plt.subplots() stats.probplot(data, dist="norm", plot=ax) ax.set_title('Normal QQ Plot') plt.show()

Common deviation patterns:

ECDF vs Normal CDF¶

Comparing the empirical CDF against the theoretical normal CDF provides a non-binned view of normality. The maximum vertical gap between the two curves is the Kolmogorov-Smirnov statistic.

```python data_sorted = np.sort(data) ecdf = np.arange(1, len(data) + 1) / len(data) cdf_normal = stats.norm.cdf(data_sorted, loc=mu, scale=sigma)

plt.step(data_sorted, ecdf, where='post', label='Empirical CDF') plt.plot(data_sorted, cdf_normal, 'r-', label='Normal CDF') plt.xlabel('Value') plt.ylabel('Cumulative Probability') plt.title('ECDF vs Normal CDF') plt.legend() plt.show()

Quantify the maximum deviation¶

ks_stat, p_value = stats.kstest(data, 'norm', args=(mu, sigma)) print(f"KS statistic: {ks_stat:.4f}, p-value: {p_value:.4f}") ```

Box Plot¶

A box plot summarizes the distribution through its quartiles and flags observations beyond the whiskers as potential outliers. For normal data, the box is approximately symmetric around the median, and few points appear beyond the whiskers.

```python fig, axes = plt.subplots(1, 2, figsize=(10, 4))

Normal data¶

axes[0].boxplot(data, vert=True) axes[0].set_title('Normal Data')

Skewed data for contrast¶

skewed = stats.lognorm.rvs(s=0.5, size=300, random_state=42) axes[1].boxplot(skewed, vert=True) axes[1].set_title('Right-Skewed Data')

plt.tight_layout() plt.show() ```

Skewness and Kurtosis Indicators¶

Quantitative measures of shape complement the visual assessment. For a normal distribution, skewness is \(0\) and excess kurtosis is \(0\).

```python skew = stats.skew(data) kurt = stats.kurtosis(data) # Excess kurtosis (Fisher) print(f"Skewness: {skew:.3f} (normal = 0)") print(f"Excess kurtosis: {kurt:.3f} (normal = 0)")

D'Agostino-Pearson test combines skewness and kurtosis¶

stat, p_value = stats.normaltest(data) print(f"D'Agostino-Pearson: statistic={stat:.3f}, p={p_value:.4f}") ```

Comparing Normal vs Non-Normal Data¶

Viewing multiple diagnostic plots side by side clarifies how departures from normality manifest across different visualization methods.

```python fig, axes = plt.subplots(2, 3, figsize=(14, 8))

for col, (label, sample) in enumerate([ ('Normal', np.random.normal(0, 1, 500)), ('Right-skewed', stats.lognorm.rvs(s=0.8, size=500, random_state=1)), ('Heavy-tailed', stats.t.rvs(df=3, size=500, random_state=2)), ]): # Histogram axes[0, col].hist(sample, bins=30, density=True, alpha=0.6) mu_fit, sigma_fit = stats.norm.fit(sample) x = np.linspace(sample.min(), sample.max(), 100) axes[0, col].plot(x, stats.norm.pdf(x, mu_fit, sigma_fit), 'r-') axes[0, col].set_title(f'{label} - Histogram')

# QQ plot
stats.probplot(sample, dist="norm", plot=axes[1, col])
axes[1, col].set_title(f'{label} - QQ Plot')

plt.tight_layout() plt.show() ```

Summary¶

Visual normality assessment reveals not just whether data are non-normal but how they deviate. Histograms show overall shape, QQ plots diagnose tail behavior and skewness with high sensitivity, and ECDF comparisons provide a complete non-binned view. Box plots highlight asymmetry and outliers at a glance, while skewness and kurtosis values quantify what the plots reveal. Using these methods together gives a thorough picture that guides the choice between parametric methods that assume normality and robust or nonparametric alternatives that do not.

Exercises¶

Exercise 1. Write code that creates a figure with three diagnostic plots for normality: histogram with normal overlay, Q-Q plot, and box plot.

Exercise 2. Explain three visual indicators of non-normality in a Q-Q plot (heavy tails, skewness, multimodality).

Exercise 3. Write code that generates data from a heavy-tailed distribution (e.g., t-distribution with 3 df) and shows its departure from normality visually.

Exercise 4. Create a function that takes a data array and produces a 4-panel normality diagnostic figure.