Although both the Logistic and Normal Distributions look very similar, there is a key difference: the Logistic Distribution has heavier tails. This means it has a higher probability of extreme values occurring further from the mean compared to the Normal Distribution.

When the scale (standard deviation) is high, the two distributions become almost identical, with the primary difference being the peak.

Visualizing the Difference Between Logistic and Normal Distributions

Here is a Python code example that plots both the normal and logistic distributions. The differences between the two can be seen visually, especially in the tails and peak of the distributions.

from numpy import random
import matplotlib.pyplot as plt
import seaborn as sns

# Plot normal and logistic distributions
sns.distplot(random.normal(scale=2, size=1000), hist=False, label='normal')
sns.distplot(random.logistic(size=1000), hist=False, label='logistic')

# Show the plot
plt.show()
      

Key Observations:

• The Logistic Distribution has more area under the tails compared to the Normal Distribution, indicating a higher probability of extreme events.
• As the scale increases, both distributions converge, but the peak of the logistic distribution becomes more pronounced.
• The logistic distribution is frequently used in machine learning models, like logistic regression, due to its suitability for modeling growth and probabilities.

Conclusion

While both the Logistic and Normal Distributions are fundamental in statistics and machine learning, their differences in the tail behavior are important when modeling real-world phenomena, especially when extreme events are more likely to occur.