Anscombe’s quartet

Anscombe’s Quartet, comprising four datasets with nearly identical summary statistics, underscores the limitations of relying solely on numerical metrics.

This article explores the quartet’s datasets, emphasizing the importance of visualizing data for a comprehensive understanding.

What is Anscombe’s Quartet?

Anscombe’s quartet comprises a set of four datasets, having identical descriptive statistical properties in terms of means, variance, R-squared, correlations, and linear regression lines but having different representations when we scatter plots on a graph.

The datasets were created by the statistician Francis Anscombe in 1973 to demonstrate the importance of visualizing data and to show that summary statistics alone can be misleading.

The four datasets that make up Anscombe’s quartet each include 11 x-y pairs of data. When plotted, each dataset seems to have a unique connection between x and y, with unique variability patterns and distinctive correlation strengths. Despite these variations, each dataset has the same summary statistics, such as the same x and y mean and variance, x and y correlation coefficient, and linear regression line.

Purpose of Anscombe’s Quartet

Anscombe’s quartet is used to illustrate the importance of exploratory data analysis and the drawbacks of depending only on summary statistics.  It also emphasizes the importance of using data visualization to spot trends, outliers, and other crucial details that might not be obvious from summary statistics alone.

Anscombe’s Quartet Dataset

The four datasets of Anscombe’s quartet.

 The above dataset can be downloaded from the link

Regression using Anscombe’s Quartet Dataset

let’s delve into the topic with practical implementation.

Import the necessary libraries

Python3

# Import the necessary libraries
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

Load the dataset

let’s import Anscombe’s quartet dataset.

Python3

df = pd.read_csv('https://query.data.world/s/6p2ntncvkzj5mnvbpkaswfilryvnrk')
print(df)

output:

    x1  x2  x3  x4     y1    y2     y3     y4
0   10  10  10   8   8.04  9.14   7.46   6.58
1    8   8   8   8   6.95  8.14   6.77   5.76
2   13  13  13   8   7.58  8.74  12.74   7.71
3    9   9   9   8   8.81  8.77   7.11   8.84
4   11  11  11   8   8.33  9.26   7.81   8.47
5   14  14  14   8   9.96  8.10   8.84   7.04
6    6   6   6   8   7.24  6.13   6.08   5.25
7    4   4   4  19   4.26  3.10   5.39  12.50
8   12  12  12   8  10.84  9.13   8.15   5.56
9    7   7   7   8   4.82  7.26   6.42   7.91
10   5   5   5   8   5.68  4.74   5.73   6.89

Find the Descriptive Statistical Properties for the all four Dataset

• Find mean for x and y for all four datasets.
• Find standard deviations for x and y for all four datasets.
• Find correlations with their corresponding pair of each datasets.
• Find slope and intercept for each datasets.
• Find R-square for each datasets.
  • To find R-square first find residual sum of square error and Total sum of square error
• Create a statistical summary by using all these data and print it. 

Python3

# mean values (x-bar)
x1_mean = df['x1'].mean()
x2_mean = df['x2'].mean()
x3_mean = df['x3'].mean()
x4_mean = df['x4'].mean()

# y-bar
y1_mean = df['y1'].mean()
y2_mean = df['y2'].mean()
y3_mean = df['y3'].mean()
y4_mean = df['y4'].mean()


# Standard deviation values (x-bar)
x1_std = df['x1'].std()
x2_std = df['x2'].std()
x3_std = df['x3'].std()
x4_std = df['x4'].std()

# Standard deviation values (y-bar)
y1_std = df['y1'].std()
y2_std = df['y2'].std()
y3_std = df['y3'].std()
y4_std = df['y4'].std()

# Correlation
correlation_x1y1 = np.corrcoef(df['x1'],df['y1'])[0,1]
correlation_x2y2 = np.corrcoef(df['x2'],df['y2'])[0,1]
correlation_x3y3 = np.corrcoef(df['x3'],df['y3'])[0,1]
correlation_x4y4 = np.corrcoef(df['x4'],df['y4'])[0,1]

# Linear Regression slope and intercept
m1,c1 = np.polyfit(df['x1'], df['y1'], 1)
m2,c2 = np.polyfit(df['x2'], df['y2'], 1)
m3,c3 = np.polyfit(df['x3'], df['y3'], 1)
m4,c4 = np.polyfit(df['x4'], df['y4'], 1)

# Residual sum of squares error
RSSY_1 = ((df['y1'] - (m1*df['x1']+c1))**2).sum()
RSSY_2 = ((df['y2'] - (m2*df['x2']+c2))**2).sum()
RSSY_3 = ((df['y3'] - (m3*df['x3']+c3))**2).sum()
RSSY_4 = ((df['y4'] - (m4*df['x4']+c4))**2).sum()

# Total sum of squares
TSS_1 = ((df['y1'] - y1_mean)**2).sum()
TSS_2 = ((df['y2'] - y2_mean)**2).sum()
TSS_3 = ((df['y3'] - y3_mean)**2).sum()
TSS_4 = ((df['y4'] - y4_mean)**2).sum()

# R squared (coefficient of determination)
R2_1  = 1 - (RSSY_1 / TSS_1)
R2_2  = 1 - (RSSY_2 / TSS_2)
R2_3  = 1 - (RSSY_3 / TSS_3)
R2_4  = 1 - (RSSY_4 / TSS_4)

# Create a pandas dataframe to represent the summary statistics
summary_stats = pd.DataFrame({'Mean_x': [x1_mean, x2_mean, x3_mean, x4_mean],
                              'Variance_x': [x1_std**2, x2_std**2, x3_std**2, x4_std**2],
                              'Mean_y': [y1_mean, y2_mean, y3_mean, y4_mean],
                              'Variance_y': [y1_std**2, y2_std**2, y3_std**2, y4_std**2],
                              'Correlation': [correlation_x1y1, correlation_x2y2, correlation_x3y3, correlation_x4y4],
                              'Linear Regression slope': [m1, m2, m3, m4],
                              'Linear Regression intercept': [c1, c2, c3, c4]},
index = ['I', 'II', 'III', 'IV'])
print(summary_stats.T)

Output:

                                     I         II        III         IV
Mean_x                        9.000000   9.000000   9.000000   9.000000
Variance_x                   11.000000  11.000000  11.000000  11.000000
Mean_y                        7.500909   7.500909   7.500000   7.500909
Variance_y                    4.127269   4.127629   4.122620   4.123249
Correlation                   0.816421   0.816237   0.816287   0.816521
Linear Regression slope       0.500091   0.500000   0.499727   0.499909
Linear Regression intercept   3.000091   3.000909   3.002455   3.001727

Clearly, we can see identical descriptive statistics summary, this uniformity in summary statistics might lead one to believe that the datasets are essentially the same.

However, when examining the scatter plots of these datasets, we’ll observe the inherent differences.

Plot the scatter plot and linear regression line for each datasets

Python3

# plot all four plots
fig, axs = plt.subplots(2, 2,  figsize=(18,12), dpi=500)

axs[0, 0].set_title('Dataset I', fontsize=20)
axs[0, 0].set_xlabel('X', fontsize=13)
axs[0, 0].set_ylabel('Y', fontsize=13)
axs[0, 0].plot(df['x1'], df['y1'], 'go')
axs[0, 0].plot(df['x1'], m1*df['x1']+c1,'r',label='Y='+str(round(m1,2))+'x +'+str(round(c1,2)))
axs[0, 0].legend(loc='best',fontsize=16)

axs[0, 1].set_title('Dataset II',fontsize=20)
axs[0, 1].set_xlabel('X', fontsize=13)
axs[0, 1].set_ylabel('Y', fontsize=13)
axs[0, 1].plot(df['x2'], df['y2'], 'go')
axs[0, 1].plot(df['x2'], m2*df['x2']+c2,'r',label='Y='+str(round(m2,2))+'x +'+str(round(c2,2)))
axs[0, 1].legend(loc='best',fontsize=16)

axs[1, 0].set_title('Dataset III',fontsize=20)
axs[1, 0].set_xlabel('X', fontsize=13)
axs[1, 0].set_ylabel('Y', fontsize=13)
axs[1, 0].plot(df['x3'], df['y3'], 'go')
axs[1, 0].plot(df['x3'], m1*df['x3']+c1,'r',label='Y='+str(round(m3,2))+'x +'+str(round(c3,2)))
axs[1, 0].legend(loc='best',fontsize=16)

axs[1, 1].set_title('Dataset IV',fontsize=20)
axs[1, 1].set_xlabel('X', fontsize=13)
axs[1, 1].set_ylabel('Y', fontsize=13)
axs[1, 1].plot(df['x4'], df['y4'], 'go')
axs[1, 1].plot(df['x4'], m4*df['x4']+c4,'r',label='Y='+str(round(m4,2))+'x +'+str(round(c4,2)))
axs[1, 1].legend(loc='best',fontsize=16)

plt.show()

Output:

Anscombe’s quartet Plot

Note: It is mentioned in the definition that Anscombe’s quartet comprises four datasets that have nearly identical simple statistical properties, yet appear very different when graphed. 

Explanation of this output:

• In the first one(top left) if you look at the scatter plot you will see that there seems to be a linear relationship between x and y.
• In the second one(top right) if you look at this figure you can conclude that there is a non-linear relationship between x and y.
• In the third one(bottom left) you can say when there is a perfect linear relationship for all the data points except one which seems to be an outlier which is indicated be far away from that line.
• Finally, the fourth one(bottom right) shows an example when one high-leverage point is enough to produce a high correlation coefficient.

Conclusion

While the descriptive statistics of Anscombe’s Quartet may appear uniform, the accompanying visualizations reveal distinct patterns, showcasing the necessity of combining statistical analysis with graphical exploration for robust data interpretation.

Anscombe Quartet – FAQs

How does Anscombe’s quartet work in a scatter plot?

Anscombe’s Quartet exhibits diverse patterns in scatter plots, illustrating the importance of visualizing data for meaningful insights beyond numerical summaries.

What are the advantages of Anscombe’s Quartet?

Reveals limitations of summary statistics, emphasizing the need for visual exploration to detect nuances, outliers, and diverse relationships in datasets.

Ho do you calculate Anscombe’s Quartet?

Analyze mean, variance, correlation, linear regression, and other metrics for each dataset within Anscombe’s Quartet to showcase identical summary statistics.

What does Anscombe’s Quartet teach us about data visualization?

Anscombe’s Quartet underscores that numerical summaries alone can be misleading, emphasizing the crucial role of data visualization in uncovering patterns and outliers.

Anscombe’s Quartet dataset in CSV?

Anscombe’s Quartet dataset can be found in CSV format at this link.