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Logistic Distribution

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Non Normal Distributions > Logistic Distribution

What is a Logistic Distribution?

By Krishnavedala (Own work) [CC0], via Wikimedia Commons

PDF of the logistic distribution. Image: Krishnavedala| Wikimedia Commons

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:
cdf normal

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:
logistic cdf

Two parameters define the shape of the distribution:

The probability density function for the distribution is:
pdf logistic distribution

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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