The sigmoid behind logistic regression, and how far it is from a normal.
How much “almost normal” is worth
The reference case. Its CDF is the sigmoid 1/(1 + e^−x) — the same function every logistic regression uses.
Logistic(μ = 0.000000, s = 1.000000)
P(X ≤ 1.000000) = 0.731059
Its standard deviation is sπ/√3 = 1.813799, not s. Against a normal with that same spread, the largest CDF difference anywhere is 0.022663 — which is how much “almost identical” is worth.
Mean = median = mode
0.000000
all equal, by symmetry
Standard deviation
1.813799
sπ/√3
Density at x
0.196612
peak is 1/(4s)
Excess kurtosis
1.2
exactly — heavier than normal
Odds at x
2.718282
p ÷ (1 − p)
Log-odds at x
1.000000
= (x − μ)/s, exactly linear
Max gap vs normal
0.022663
at -0.683 sd
Tail ratio at 3 sd
3.19633
times the normal's
Logistic minus normal, variance-matched
The difference is never more than 2.2663percentage points, and its extremes sit at about ±0.68 standard deviations — near the shoulders, not in the tails. Out in the tails the RATIO diverges even though the difference shrinks, which is why the two look interchangeable on a plot and are not for anything that depends on extremes.
The log-odds is exactly linear in x. log(p ÷ (1 − p)) = (x − μ) ÷ s, with no approximation anywhere. That is the entire reason logistic regression is built on this distribution rather than the normal one: it makes the model linear in the parameters on the log-odds scale, and it is why a coefficient there is read as a log odds ratio rather than as a change in probability.
Matched on variance, the two CDFs never differ by more than 0.022663. That is small enough that the choice between a probit and a logit model rarely changes a conclusion, and large enough to matter when the question is about a tail. At three standard deviations the logistic puts 3.1963 times as much probability beyond the point as the normal does, and the ratio grows from there.
Its standard deviation is sπ/√3, not s. A scale of 1 gives a standard deviation of 1.813799, so comparing a logistic scale directly against a normal sigma overstates the spread by 81%. The variance-matched preset sets s = √3/π = 0.5513 for exactly this reason.
What this tool shows
Matched on variance, the logistic and normal CDFs never differ by more than 0.022663 — and that maximum sits at ±0.68 standard deviations, not in the tails. Out in the tails the difference shrinks while the RATIO grows: at three standard deviations the logistic puts 3.20 times as much probability beyond the point. The tool computes both.
PDF, CDF, odds and log-odds for any location and scale
The exact maximum difference from a variance-matched normal, and where it occurs
The tail ratio at three standard deviations
The scale-to-standard-deviation conversion, sπ/√3 — a common slip
The log-odds being exactly linear in x, which is why logistic regression uses it
The difference curve plotted across six standard deviations
The sigmoid Exact log-odds Gap measured Tail ratio
Updated 12 September 2026 · Works in any browser, no installation
The logistic distribution’s CDF is the sigmoid 1/(1 + e⁻ᵙ) — the same function every logistic regression uses to turn a linear predictor into a probability. It looks like a normal distribution with slightly heavier tails, and its log-odds is exactly linear in x, which is the reason it is the model rather than the normal.
At a glance
Formula shown
The CDF is F(x) = 1/(1 + e^(−(x−μ)/s)) and the density f(x) = e^(−z)/(s(1 + e^(−z))²) with z = (x−μ)/s. The mean, median and mode all equal μ. The variance is s²π²/3, so the standard deviation is sπ/√3 ≈ 1.8138s — not s. Excess kurtosis is exactly 1.2. The log-odds, log(p/(1−p)), equals (x − μ)/s exactly, with no approximation anywhere.
Scenario support
Logistic regression and the logit link, growth and adoption S-curves, item response theory in educational testing, choice models in economics, and any place a probability has to be modelled as a smooth function of a linear predictor.
Educational estimate
Planning support from the values you enter — not professional advice.
How close to a normal, exactly
The two are described as nearly interchangeable. Putting them on the same variance and measuring gives a number rather than an impression.
The largest difference between the two CDFs is 0.022663. Just over two percentage points, anywhere on the real line, once the scales are matched.
And it occurs at ±0.68 standard deviations, near the shoulders rather than in the tails — which is the opposite of where people expect two distributions to disagree.
In the tails the DIFFERENCE shrinks while the RATIO grows. At three standard deviations the absolute gap is small and the logistic puts 3.1963 times as much probability beyond the point. Both facts are true, and which one matters depends entirely on the question.
So a probit and a logit model rarely disagree about a conclusion — the fitted probabilities differ by at most a couple of points — and they can disagree substantially about a rare event.
The tool plots the whole difference curve across six standard deviations, so the shape of the disagreement is visible rather than summarised by its maximum.
The log-odds is exactly linear
This is the property that made the logistic distribution the basis of a whole family of models, and it is exact rather than approximate.
log(p ÷ (1 − p)) = (x − μ) ÷ s. Invert the sigmoid and the algebra collapses to a straight line. No series expansion, no numerical inversion.
Which makes a regression linear in its parameters on the log-odds scale. That is the entire reason logistic regression uses this link: the model is linear where it needs to be, and the fitted values are automatically probabilities.
And it is why a coefficient is a log odds ratio. A one-unit increase in a predictor adds its coefficient to the log-odds, which multiplies the odds by e raised to it. Not the probability — the odds, which is the reading people most often get wrong.
The probit has no such closed form. A normal CDF does not invert in elementary functions, so the equivalent model needs numerical machinery throughout. That practical difference, not a statistical one, is why the logit dominates.
The scale is not the standard deviation
The commonest error with this distribution is a units error, and it inflates or deflates everything downstream.
The standard deviation is sπ/√3 ≈ 1.8138s. A scale of 1 gives a spread of 1.8138, so comparing a logistic scale directly against a normal sigma understates the spread by 81%.
The variance-matched preset sets s = √3/π = 0.5513. That is the scale at which the comparison with a standard normal is fair, and it is the setting the difference figures on this page are computed at.
The density peaks at 1/(4s), which is another place the two parameterisations diverge visually: a logistic and a normal with the same s look very different, and with the same standard deviation look almost identical.
And the excess kurtosis is exactly 1.2, a rational number rather than an approximation — one of the few distributions where that summary comes out clean, and a quick way to confirm you are looking at a logistic rather than something fitted to resemble one.
The S-curve, and what it assumes
The same CDF is the standard model for growth that starts slowly, accelerates and then saturates.
The curve is symmetric about its midpoint. Growth accelerates to the halfway point and decelerates identically afterwards. That symmetry is an assumption, not an observation, and real adoption curves are often asymmetric.
The scale s sets how long the transition takes. Roughly, the curve moves from 10% to 90% over about 4.4s, which is a more useful summary than the maximum slope for anyone planning against it.
And the ceiling is assumed, not fitted. The logistic CDF saturates at 1; a growth model has to supply the real ceiling, and the fitted curve is extremely sensitive to it. Most disagreements about an S-curve forecast are disagreements about the ceiling rather than about the shape.
Extrapolating before the inflection point is unreliable. Early exponential-looking growth is consistent with many ceilings, so an S-curve fitted to data that has not yet turned over carries most of its information in the assumption rather than in the data.
Where it is the model, not an approximation
Three places the logistic distribution is chosen deliberately rather than as a stand-in for a normal.
Discrete choice models in economics. If the unobserved part of a decision-maker’s utility follows an extreme-value distribution, the DIFFERENCE of two such terms is logistic — which derives the logit model from a behavioural assumption rather than by convenience.
Item response theory. The probability of a correct answer as a function of ability is modelled with this curve, with the location as item difficulty and the scale’s reciprocal as discrimination. The whole of modern educational testing rests on it.
Neural network activations. The sigmoid that squashed activations for decades is this CDF, and the softmax that generalises it to several classes is the multinomial version of the same model.
And it is the limiting distribution of a logistic growth process, which is where the name comes from — Verhulst’s population model, long before it became a statistical distribution.
Reporting logistic results
Four things, and the second is where most logistic regression write-ups go wrong.
State the scale convention. Some software parameterises by the standard deviation rather than s, and the two differ by a factor of 1.81.
Do not read a coefficient as a change in probability. It is a change in log-odds. The probability change depends on where you start — largest near 50%, almost nothing near 0 or 100%.
Quote a marginal effect if probability is what you mean. The derivative of the probability with respect to the predictor, evaluated somewhere stated, is the number most readers assume a coefficient already is.
And say whether it was a logit or a probit. The coefficients differ by roughly the 1.81 factor and are not comparable across the two, even though the fitted probabilities are nearly identical.
Sources and methodology
References for the logistic distribution and the logit link.
Method. The CDF and density are the exact closed forms, and the comparison against a normal is computed rather than quoted: the maximum difference is found by scanning 8,001 points across six standard deviations after matching the variances with s = √3/π. The suite pins that maximum at 0.022663, confirms it occurs within 1.2 standard deviations of the centre rather than in the tails, and pins the three-sigma tail ratio at 3.1963. The density is confirmed to integrate to its own CDF range by Simpson’s rule and the CDF to be monotone across 400 points. That engine is verified on every change against 75 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Normal DistributionProbabilities under a normal curve in all four directions with the region shaded — and the empirical rule given exactly, because two standard deviations is 95.45% and the 95% everyone quotes sits at 1.96σ.
Odds RatioOdds ratio, relative risk, risk difference and number needed to treat from one 2x2 table — because an odds ratio of 6.00 can describe a relative risk of 1.50.
Laplace DistributionFits by maximum likelihood and shows the location estimate IS the sample median, with the log-likelihood at the mean beside it — the fact that least-absolute-deviations regression is built on.
Exponential DistributionTime-until-event probabilities from a rate or a mean, always showing both — with a panel proving elapsed time changes nothing, which is a constant hazard rate and false for anything that wears out.
Sensitivity and SpecificitySensitivity, specificity, PPV, NPV, likelihood ratios and MCC from a 2×2 table, with predictive values recomputed across the prevalence range — a 99%/99% test has a PPV of 50% at 1% prevalence and 9% at 0.1%.
Relative RiskRisk ratio and odds ratio from one table with the divergence between them plotted: they agree to half a percent at a 1% baseline, and at an 80% baseline the odds ratio is exactly half the risk ratio.
An educational tool. The logistic and normal distributions are close enough that the choice between a logit and a probit rarely changes a conclusion about a typical case — and far enough apart in the tails that it can change a conclusion about a rare one, which is where the choice should be made deliberately.
Published the logistic distribution with a number attached to 'almost the same as a normal': matched on variance the two CDFs never differ by more than 0.022663, found by scanning 8,001 points across six standard deviations.
That maximum occurs at plus or minus 0.68 standard deviations - near the shoulders, not in the tails, which is the opposite of where people expect two distributions to disagree.
In the tails the difference shrinks while the ratio grows: at three standard deviations the logistic puts 3.1963 times as much probability beyond the point. Both facts are true and which matters depends on the question.
The log-odds is exactly linear in x with no approximation anywhere, which is the entire reason logistic regression is built on this distribution rather than the normal - and why a coefficient there is a log odds ratio rather than a change in probability.
The scale-to-standard-deviation conversion is s*pi/sqrt(3), not s; comparing a logistic scale directly against a normal sigma understates the spread by 81%.
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