Q-Q Plot (Quantile-Quantile Plot)

Purpose

Check if a variable follows a normal distribution by comparing its quantiles to theoretical normal distribution quantiles.

Analysis Type

Univariate

What to Look For

1. Normal Distribution (GOOD)

• Points fall on reference line: Data is normally distributed
• Small deviations are acceptable
• Important for many statistical tests and ML algorithms

2. Heavy Tails

• Points curve above line at ends: More extreme values than normal
• Distribution has fatter tails

3. Light Tails

• Points curve below line at ends: Fewer extreme values than normal
• Distribution has thinner tails

4. Right Skew (Positive Skew)

• Points curve above line on right: Long right tail
• Most values concentrated on left
• Solution: Apply log or square root transformation

5. Left Skew (Negative Skew)

• Points curve above line on left: Long left tail
• Most values concentrated on right
• Solution: Apply power transformation

6. S-Shape:

• Heavy tails on both ends
• May indicate mixture of distributions

7. Gaussian distribution

• Points on the diagonal line: Data is normally distributed
• S-shaped curve: Heavy tails (more extreme values than normal)
• Inverted S-curve: Light tails (fewer extreme values)
• Points curve upward on right: Right-skewed distribution
• Points curve downward on left: Left-skewed distribution

Code Example

from scipy import stats
import statsmodels.api as sm

# Load the longley dataset
data = sm.datasets.longley.load_pandas()
data = data.exog  # Use explanatory variables

# Q-Q plot
stats.probplot(data['GNP'], dist="norm", plot=plt)
plt.title("Q-Q Plot (with scipy)")
plt.show()

# Using statsmodels
sm.qqplot(data['GNP'], line='s')
plt.title("Q-Q Plot (with statsmodels)")
plt.show()