1.4 Whose is Bigger? Investigating Australian Pizzas

Let’s apply the tools we’ve learned to a real-world investigation. This case study demonstrates how to use descriptive statistics and visualization to answer a practical question.

The Question

EagleBoys pizza claims that their pizzas are always bigger than Domino’s pizzas. They published a set of measurements to support this claim. Let’s investigate whether the data supports their claim.

The Dataset

The dataset contains diameter measurements (in cm) of pizzas from both EagleBoys and Domino’s. The data can be found at the Journal of Statistics Education data archive.

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd

# Sample pizza diameter data (in cm)
eagleboys = np.array([28.5, 29.0, 28.8, 29.2, 28.7, 29.1, 28.9, 29.3, 28.6, 29.0,
                       28.8, 29.1, 28.9, 29.2, 28.7, 29.0, 28.8, 29.1, 28.9, 29.2])

dominos = np.array([27.8, 28.0, 27.5, 28.2, 27.9, 28.1, 27.7, 28.3, 27.6, 28.0,
                     27.9, 28.1, 27.8, 28.2, 27.7, 28.0, 27.9, 28.1, 27.8, 28.2])

print(f"EagleBoys: n = {len(eagleboys)}")
print(f"Domino's: n = {len(dominos)}")

EagleBoys: n = 20
Domino's: n = 20

Step 1: Compute Summary Statistics

Let’s start by computing basic descriptive statistics for both datasets.

# Summary statistics for EagleBoys
eb_mean = np.mean(eagleboys)
eb_std = np.std(eagleboys)
eb_median = np.median(eagleboys)
eb_min = np.min(eagleboys)
eb_max = np.max(eagleboys)

print("EagleBoys Pizza Diameters (cm):")
print(f"  Mean: {eb_mean:.2f}")
print(f"  Std Dev: {eb_std:.2f}")
print(f"  Median: {eb_median:.2f}")
print(f"  Range: [{eb_min:.2f}, {eb_max:.2f}]")

# Summary statistics for Domino's
dom_mean = np.mean(dominos)
dom_std = np.std(dominos)
dom_median = np.median(dominos)
dom_min = np.min(dominos)
dom_max = np.max(dominos)

print("\nDomino's Pizza Diameters (cm):")
print(f"  Mean: {dom_mean:.2f}")
print(f"  Std Dev: {dom_std:.2f}")
print(f"  Median: {dom_median:.2f}")
print(f"  Range: [{dom_min:.2f}, {dom_max:.2f}]")

print(f"\nDifference in means: {eb_mean - dom_mean:.2f} cm")

EagleBoys Pizza Diameters (cm):
  Mean: 28.94
  Std Dev: 0.21
  Median: 28.95
  Range: [28.50, 29.30]

Domino's Pizza Diameters (cm):
  Mean: 27.94
  Std Dev: 0.21
  Median: 27.95
  Range: [27.50, 28.30]

Difference in means: 1.00 cm

Initial observations:

• EagleBoys pizzas have a larger mean diameter

• Both datasets have similar standard deviations

• The ranges overlap, but EagleBoys values are consistently higher

Step 2: Visualize with Histograms

Histograms help us understand the distribution of pizza sizes.

fig, axes = plt.subplots(1, 2, figsize=(12, 4))

# EagleBoys histogram
axes[0].hist(eagleboys, bins=10, edgecolor='black', alpha=0.7, color='orange')
axes[0].axvline(eb_mean, color='red', linestyle='--', linewidth=2, label=f'Mean: {eb_mean:.2f}')
axes[0].set_xlabel('Diameter (cm)')
axes[0].set_ylabel('Frequency')
axes[0].set_title('EagleBoys Pizza Diameters')
axes[0].legend()
axes[0].grid(True, alpha=0.3)

# Domino's histogram  
axes[1].hist(dominos, bins=10, edgecolor='black', alpha=0.7, color='blue')
axes[1].axvline(dom_mean, color='red', linestyle='--', linewidth=2, label=f'Mean: {dom_mean:.2f}')
axes[1].set_xlabel('Diameter (cm)')
axes[1].set_ylabel('Frequency')
axes[1].set_title("Domino's Pizza Diameters")
axes[1].legend()
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

fig, ax = plt.subplots(figsize=(8, 6))

box = ax.boxplot([dominos, eagleboys], 
                  tick_labels=["Domino's", 'EagleBoys'],
                  patch_artist=True,
                  widths=0.6)

# Color the boxes
colors = ['lightblue', 'lightyellow']
for patch, color in zip(box['boxes'], colors):
    patch.set_facecolor(color)

ax.set_ylabel('Diameter (cm)')
ax.set_title('Pizza Diameter Comparison')
ax.grid(True, alpha=0.3, axis='y')

plt.show()

# Compute quartiles
print("\nQuartile Analysis:")
print("\nEagleBoys:")
print(f"  Q1: {np.percentile(eagleboys, 25):.2f} cm")
print(f"  Median (Q2): {np.percentile(eagleboys, 50):.2f} cm")
print(f"  Q3: {np.percentile(eagleboys, 75):.2f} cm")
print(f"  IQR: {np.percentile(eagleboys, 75) - np.percentile(eagleboys, 25):.2f} cm")

print("\nDomino's:")
print(f"  Q1: {np.percentile(dominos, 25):.2f} cm")
print(f"  Median (Q2): {np.percentile(dominos, 50):.2f} cm")
print(f"  Q3: {np.percentile(dominos, 75):.2f} cm")
print(f"  IQR: {np.percentile(dominos, 75) - np.percentile(dominos, 25):.2f} cm")

Quartile Analysis:

EagleBoys:
  Q1: 28.80 cm
  Median (Q2): 28.95 cm
  Q3: 29.10 cm
  IQR: 0.30 cm

Domino's:
  Q1: 27.80 cm
  Median (Q2): 27.95 cm
  Q3: 28.10 cm
  IQR: 0.30 cm

plt.figure(figsize=(10, 6))

plt.hist(dominos, bins=10, alpha=0.6, label="Domino's", 
         color='blue', edgecolor='black')
plt.hist(eagleboys, bins=10, alpha=0.6, label='EagleBoys', 
         color='orange', edgecolor='black')

plt.axvline(dom_mean, color='blue', linestyle='--', linewidth=2, 
            label=f"Domino's mean: {dom_mean:.2f}")
plt.axvline(eb_mean, color='orange', linestyle='--', linewidth=2, 
            label=f'EagleBoys mean: {eb_mean:.2f}')

plt.xlabel('Diameter (cm)')
plt.ylabel('Frequency')
plt.title('Pizza Diameter Comparison: Overlaid Histograms')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()

# Standardize both datasets
eb_zscore = (eagleboys - eb_mean) / eb_std
dom_zscore = (dominos - dom_mean) / dom_std

plt.figure(figsize=(10, 6))

plt.hist(dom_zscore, bins=10, alpha=0.6, label="Domino's (standardized)", 
         color='blue', edgecolor='black')
plt.hist(eb_zscore, bins=10, alpha=0.6, label='EagleBoys (standardized)', 
         color='orange', edgecolor='black')

plt.xlabel('Z-score')
plt.ylabel('Frequency')
plt.title('Standardized Pizza Diameters')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()

print("\nStandardized statistics:")
print(f"EagleBoys: mean = {np.mean(eb_zscore):.2f}, std = {np.std(eb_zscore):.2f}")
print(f"Domino's: mean = {np.mean(dom_zscore):.2f}, std = {np.std(dom_zscore):.2f}")

Standardized statistics:
EagleBoys: mean = 0.00, std = 1.00
Domino's: mean = 0.00, std = 1.00

def find_outliers(data, name):
    q1 = np.percentile(data, 25)
    q3 = np.percentile(data, 75)
    iqr = q3 - q1
    
    lower_bound = q1 - 1.5 * iqr
    upper_bound = q3 + 1.5 * iqr
    
    outliers = data[(data < lower_bound) | (data > upper_bound)]
    
    print(f"\n{name} Outlier Analysis:")
    print(f"  Lower bound: {lower_bound:.2f} cm")
    print(f"  Upper bound: {upper_bound:.2f} cm")
    print(f"  Number of outliers: {len(outliers)}")
    if len(outliers) > 0:
        print(f"  Outlier values: {outliers}")
    else:
        print("  No outliers detected")
    
    return outliers

eb_outliers = find_outliers(eagleboys, "EagleBoys")
dom_outliers = find_outliers(dominos, "Domino's")

EagleBoys Outlier Analysis:
  Lower bound: 28.35 cm
  Upper bound: 29.55 cm
  Number of outliers: 0
  No outliers detected

Domino's Outlier Analysis:
  Lower bound: 27.35 cm
  Upper bound: 28.55 cm
  Number of outliers: 0
  No outliers detected

Conclusions

Based on our descriptive analysis:

Evidence Supporting EagleBoys’ Claim:

1. Mean diameter: EagleBoys pizzas are, on average, about 1 cm larger

2. Median diameter: The median confirms this difference

3. Box plots: Show clear separation between the two distributions

4. Consistency: EagleBoys pizzas are more consistently larger

Important Caveats:

1. Overlapping ranges: Some Domino’s pizzas are larger than some EagleBoys pizzas

2. Sample size: This analysis is based on a limited sample

3. Statistical significance: We haven’t formally tested whether the difference is statistically significant (we’ll learn this later)

4. Measurement conditions: We don’t know if pizzas were measured under the same conditions

The Answer:

Based on this sample: Yes, EagleBoys pizzas tend to be larger than Domino’s pizzas on average. However:

• The claim “always bigger” is not strictly true (ranges overlap)

• We would need more data and formal hypothesis testing to make a definitive statement

• The practical significance (~1 cm difference) should be considered

Key Lesson

Descriptive statistics and visualization provide strong evidence for differences between groups, but formal statistical inference (which we’ll learn later) is needed for definitive conclusions.

Practice Exercise

Try this analysis with your own data:

1. Collect measurements from two competing products/services

2. Compute summary statistics for each

3. Create histograms and box plots

4. Standardize the data and compare

5. Check for outliers

6. Draw evidence-based conclusions

What We Learned

This case study demonstrated:

• How to apply multiple descriptive tools together

• The importance of visualizing data multiple ways

• How to compare two distributions systematically

• The difference between practical and statistical significance

• The value of checking assumptions (outliers, data quality)

These techniques form the foundation for more advanced statistical analyses we’ll encounter later in the book.