Logistic Distribution Calculator

How it is calculated

The tool standardises your point, z = (x - μ) / s, where μ is the location (the centre of the curve) and s is the scale (how spread out it is; it must be above zero, otherwise there is no result). It then evaluates the cumulative probability F(x) = 1 / (1 + e^(-z)), the density f(x) = e^(-z) / (s × (1 + e^(-z))²), the survival probability 1 - F(x), the mean (equal to μ) and the standard deviation s × π / √3. The formulas are arranged so that very large |z| does not overflow: a point thousands of scale units from μ simply shows 0.000000 or 1.000000. Probabilities and density are shown to six decimals, the mean to two and the standard deviation to four. The density is the height of the curve, not a probability, and can exceed 1 when s is small.

Common mistakes

• Treating the scale s as the standard deviation. For the logistic distribution the standard deviation is s × π / √3, about 1.81 times s.
• Reading the density as a probability. It is the height of the curve at x; probabilities come from the cumulative value or differences of it.
• Entering a zero or negative scale. s must be greater than zero; the tool gives no result otherwise.
• Using normal-distribution tables for a logistic problem. The two look alike, but the logistic has heavier tails, so tail probabilities differ.
• Confusing this distribution with the logistic function in regression. That function is the cumulative curve with μ = 0 and s = 1; fitted regression coefficients are not the μ and s used here.