If $Y$ is lognormally distributed, what is $E|Y - 1|$? Let $Y$ be lognormal with parameters $\mu$ and $\sigma$, such that $\log(Y)$ is Gaussian with mean $\mu$ and variance $\sigma^2$.
I know $0 < Y < \infty$, $E(Y) = \exp(\mu+\frac{1}{2}\sigma^2)$, and $Var(Y) = \exp(2\mu +\sigma^{2})[\exp(\sigma^{2})-1]$
Let $R = Y - 1$.
$-1 < R < \infty$, $E(R) = \exp(\mu+\frac{1}{2}\sigma^2) - 1$, and $Var(R) = Var(Y) = \exp(2\mu +\sigma^{2})[\exp(\sigma^{2})-1]$
What is $E(|R|)$?
This is the mean of the "folded" distribution of $R$.
Context:
When you compare a measurement $M$ to a known control $C$, the quantity $\frac{M}{C}$ is often lognormally distributed.  $\frac{M-C}{C} = \frac{M}{C} - 1$ is the "relative difference" or RD.  In the above, $\frac{M}{C} = Y$ and $Y - 1 = R$ is the RD.  $E(R)$ is the "mean relative difference" or MRD.
$|R|$ is the "absolute relative difference" (not my choice of terms) or ARD, and $E(|R|)$ is the "mean absolute relative difference" or MARD.  This quantity is used to summarize the accuracy of a measurement system, especially the accuracy of blood glucose monitors.

According to Wolfram Mathematica, in general, for $a>0$, $\mathbb{E}_{x\sim\mathrm{N}(\mu,\sigma^2)}[|e^x - a|] $ is given by
$$
e^{\mu+\frac{\sigma^2}{2}}\,
\mathrm{Erf}\bigg(\frac{\mu+\sigma^2-\log a}{\sqrt{2\sigma^2}}\bigg) -
a\,
\mathrm{Erf}\bigg(\frac{\mu-\log a}{\sqrt{2\sigma^2}}\bigg).
$$
