Logistic Distribution

Non Normal Distributions > Logistic Distribution

What is a Logistic Distribution?

The logistic distribution is used for modeling growth, and also for logistic regression. It is a symmetrical distribution, unimodal (it has one peak) and is similar in shape to the normal distribution. In fact, the logistic and normal distributions are so close in shape (although the logistic tends to have slightly fatter tails) that for most applications it’s impossible to tell one from the other.

Why is it Used?

The logistic distribution is mainly used because the curve has a relatively simple cumulative distribution formula to work with. The formula approximates the normal distribution extremely well.

Finding cumulative probabilities for the normal distribution usually involves looking up values in the z-table, rounding up or down to the nearest z-score. Exact values are usually found with statistical software, because the cumulative distribution function is so difficult to work with, involving integration:



Although there are many other functions that can approximate the normal, they also tend to have very complicated mathematical formulas. The logistic distribution, in comparison, has a much simpler CDF formula:



Two parameters define the shape of the distribution:

• The location parameter (μ) tells you where it’s centered on the x-axis.
• The scale parameter (σ) tells you what the spread is. In the above equation, s is a scale parameter proportional to the standard deviation.

The probability density function for the distribution is:

Other Useful Statistics

Mean (μ) = m
Variance (σ)² = ⅓ π² b²
Skewness (γ)₁ = 0
Kurtosis (γ)₂ = ⁶/₅

Fun fact: The United States Chess Federation and the World Chess Federation (FIDE) use the logistic distribution to calculate the relative skill level of chess players. Formerly, they both used the normal distribution.

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