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The standard Gumbel distribution for maxima has distribution function $\Lambda (z)=e^{-e^{-z}}$ for every real $z$. Its probability density function is the derivative of its distribution function:This article uses the maxima convention throughout. Reflecting a standard Gumbel distribution random variable $Z$ to $-Z$ gives the minima convention, with distribution function $1-\exp (-e^z)$. The two conventions have opposite directions of asymmetry, so their formulas must not be interchanged. ^[ref].

For a location $\mu \in \mathbb{R}$ and positive scale $b>0$, let $Y=\mu +bZ$. This location-scale family has cumulative distribution function and probability density functionSolving $\Lambda (z)=p$ gives the standard quantile function $q(p)=-\log (-\log p)$, $0<p<1$, and hence the standard medianThe median of $Y$ is therefore $\mu -b\log (\log 2)$. Location $\mu$ is not the median: at $x=\mu$ the cumulative distribution function is $e^{-1}\approx 0.367879$.

To derive the Gumbel distribution as an extreme-value limit, take the maximum of independent exponential random variables with rate $\lambda >0$ and setThe exact distribution function isThe finite-$n$ lower boundary is required because $M_n\ge 0$. For each fixed real $z$, eventually $z>-\log n$, and the natural logarithm of the first branch satisfiesExponentiating proves convergence in distribution to the standard Gumbel distribution. It is the familiar limit $(1-u/n{)}^n\to e^{-u}$ with $u=e^{-z}$: a simple calculation reveals a whole limiting shape. The name honors Emil Gumbel; Siegrist gives this exponential-maxima derivation and the standard Gumbel distribution formulas. ^[ref].

For the toy delay model $\lambda =1{\mathrm{s}}^{-1}$, $n=100$, the limiting Gumbel distribution approximation to $M_{100}$ has $\mu =(\log 100)\mathrm{s}$ and $b=1\mathrm{s}$. Its predicted median iscompared with the exact median $-\log (1-2^{-1/100})\mathrm{s}\approx 4.975147\mathrm{s}$. The approximation is about $0.003464$ seconds lower in this example. It is not an exact finite-sample probability distribution: the approximating Gumbel distribution assigns positive probability to negative delays, while the exact sample maximum cannot be negative.

The maximum domain of attraction of the Gumbel distribution is characterized by an exponential tail ratio on the scale of a Gumbel auxiliary function. This allows many original probability distributions to share the limiting shape, but it does not make the centering $\log n$ appropriate to all of them.

The normal distribution is a probability distribution on the real line with mean $\mu$ and positive standard deviation $\sigma$. Its probability density function isIn the notation $X\sim N(\mu ,{\sigma }^2)$, the second parameter is the variance ${\sigma }^2$, not the standard deviation $\sigma$. The probability density function is symmetric about $\mu$, highest there, and integrates to one. These formulas and the parameter convention are given in MIT 18.05, Reading 5c, §5: normal distributions.

The normal distribution is a location-scale family: if $Z$ is a standard normal random variable, then $X=\mu +\sigma Z$. Changing $\mu$ moves the center; changing $\sigma$ stretches the horizontal distances from it. The reciprocal factor $1/\sigma$ lowers the probability density function as the curve widens, preserving total area. This is why one standard normal distribution suffices to calculate probabilities for the whole family. ^[ref].

The height $f_X(x)$ is a probability density function value, not $\mathbb{P}(X=x)$. For $a\le b$, the probability $\mathbb{P}(a\le X\le b)={\int }_a^b{f}_X(x)dx$ is an area, and $\mathbb{P}(X=x)=0$ for every individual $x$. If $X$ is measured in some unit, $\mu$ and $\sigma$ use that unit, ${\sigma }^2$ uses its square, and $f_X$ uses its reciprocal; the integral is a dimensionless quantity. A change of measurement unit therefore changes the height without changing the probability. The area interpretation is developed in MIT 18.05, Reading 5b: continuous random variables.

The usefulness of the normal distribution goes beyond its simple shape. Under the central limit theorem, sums of independent and identically distributed random variables with finite positive variance, after centering and scaling, approach the standard normal distribution. Thus approximations by a normal distribution can arise even when the individual random variables have quite different probability distributions. The hypotheses matter: a model of measurement error still needs evidence that this approximation is appropriate. ^[ref]. Exact closure under a linear combination of independent normal random variables provides another reason the family is convenient.

Historically, the curve appeared in a different problem from fitting measurements. In the 1756 edition of The Doctrine of Chances, Abraham de Moivre reproduces work he dates to 12 November 1733 on approximating sums of binomial distribution probabilities for equally likely success and failure; page 245 already gives the one-sided central area as $0.341344$. ^[ref]. In his 1809 Theoria motus, Carl Friedrich Gauss derived a normal distribution for measurement error from a condition on the arithmetic mean, then connected maximizing its likelihood to minimizing the sum of squared errors under that model, the principle behind ordinary least squares. These are distinct contributions: approximating binomial distribution probabilities and justifying an estimation rule. ^[ref].

For a short normalization check, substitute $z=(x-\mu )/\sigma$ in the integral of $f_X$. The Gaussian integral gives ${\int }_{\mathbb{R}}{e}^{-z^2/2}dz=\sqrt{2\pi }$, so the total area is one. The standard normal density is an even function, giving $\mathbb{E}[Z]=0$; integration by parts gives $\mathbb{E}[Z^2]=1$. Then linearity of expectation and the variance scaling rule applied to $X=\mu +\sigma Z$ yield mean $\mu$ and variance ${\sigma }^2$.

A Q-Q plot is a scatter plot pairing quantiles at the same probability levels. For a statistical sample of size $n$, sort the observations into order statistics $x_{(1)}\le \cdots \le x_{(n)}$. Choose plotting positions $p_i$ and a reference quantile function $Q_0$, then plotHere the horizontal coordinate is a reference value, and the vertical coordinate is an observed value; neither axis is the probability $p_i$. The displayed plotting positions are our chosen convention, placing every level strictly between zero and one. They are not a claim that the $i$th order statistic equals that population quantile. For two statistical samples, use sample quantiles at matching levels, with a stated interpolation convention when their sizes differ. ^[ref]

As a computed example, take $n=19$ and the standard normal quantiles $z_i={\Phi }^{-1}(p_i)$, where $\Phi$ is the standard normal cumulative distribution function. Construct the ordered values $a_i=10+2z_i$. At ranks $1,10,19$, the plotting positions are $1/38,1/2,37/38$, the reference quantiles are approximately $-1.937932,0,1.937932$, and the constructed values are approximately $6.124137,10,13.875863$. Every point lies on $a=10+2z$ by construction. These 19 values are deliberately chosen, not random draws or real measurements.

The usefulness of a Q-Q plot is that it turns a comparison of probability distributions into geometry: if a reference random variable $Z$ is transformed into $X=\mu +\sigma Z$ with $\sigma >0$, its quantile function satisfies $Q_X(p)=\mu +\sigma Q_Z(p)$. This follows because a positive scaling preserves ordering. Thus the location-scale family produces a straight population Q-Q plot, and an approximate sample line suggests a compatible shape over the observed range. Its intercept describes location and its slope describes scale. The identity line requires the same location, scale and units on both axes; even then its displayed angle is 45 degrees only with equal axis scales. Curvature is often more informative than a histogram because it shows which quantiles depart from the reference. ^[ref]