Calculating grounded emitter amplifer distortion (Art of Electronics 3rd Ed.)
The problem occurs because \$r_e=\frac{k\:T}{q\:I_\text{E}}\$ and because the gain is \$A_V=\frac{R_\text{C}}{r_e}\$ with the emitter grounded. Ignoring the base's recombination current, \$I_\text{E}\approx I_\text{C}=\frac{V_\text{CC}-V_{\text{C}}}{R_\text{C}}\$, so:

$$A_V=\frac{R_\text{C}}{\frac{k\:T}{q\:\frac{V_\text{CC}-V_{\text{C}}}{R_\text{C}}}}=\frac{R_\text{C}}{\frac{k\:T}{q}}\cdot\frac{V_\text{CC}-V_{\text{C}}}{R_\text{C}}=\frac{V_\text{CC}-V_{\text{C}}}{\frac{k\:T}{q}}$$

Note that the collector resistor disappeared. Also, you should know that \$V_T=\frac{k\:T}{q}\approx 26\:\text{mV}\$ at room temperature. So the above equation, discounting temperature variations, becomes:

$$A_V=\frac{V_\text{CC}-V_{\text{C}}}{26\:\text{mV}}\tag{1}\label{eq1}$$

In your example, \$V_\text{CC}=10\:\text{V}\$ and the nominal \$V_\text{C}=5\:\text{V}\$, so with the signal value at its midpoint the gain is about \$A_V\approx 192.31\$. But with \$V_\text{C}=5\:\text{V}+100\:\text{mV}=5.1\:\text{V}\$ then \$A_V\approx 188.46\$ and with \$V_\text{C}=5\:\text{V}-100\:\text{mV}=4.9\:\text{V}\$ then \$A_V\approx 196.15\$. Roughly, this is about a gain change of \$\pm 4\$ riding on an average gain of about \$200\$ (rounding things up a bit.) That works out to about \$\pm 2\:\%\$.

But this is the "peak" figure, as the text mentions. The text suggests that you divide this value by 3, as a rule they provide, to get the "waveform distortion." This computes out to \$ 0.6\overline{6}\:\%\$ and they round it to \$\pm 0.7\:\%\$.

(I suspect they mean by this to imply total harmonic distortion. If so, the calculation for this is complex and beyond the scope of my answer in order to prove their rule of thumb answer. But there are also several different meanings for distortion, too. And a complete comparison of them here would be far, far beyond my desire. So you will simply have to accept their approach for these purposes. They have the tools on hand for the analysis and were able to empirically test their estimates. So let's leave it at that.)

It's simple enough for you to use the above formula, using \$V_\text{C}=5\:\text{V}-1\:\text{V}=4\:\text{V}\$ and also \$V_\text{C}=5\:\text{V}+1\:\text{V}=6\:\text{V}\$ to work out, after dividing by 3 again, their figure of \$\approx 6.6\:\%\$.

Again assuming that \$I_\text{C}\approx I_\text{E}\$ and using an emitter resistor to degrade the gain (intentionally, often quite substantially), equation \$\ref{eq1}\$ becomes:

$$A_V=\frac{1}{\frac{26\:\text{mV}}{V_\text{CC}-V_\text{C}}+\frac{R_\text{E}}{R_\text{C}}}\tag{2}\label{eq2}$$

(If you need a walk-through to see how that equation arrives, I can provide it. I'm leaving it this way so that you have to do some work to see if you can arrive at the same place on your own.)

So long as \$\frac{R_\text{E}}{R_\text{C}}\gg \frac{26\:\text{mV}}{V_\text{CC}-V_\text{C}}\$, for all \$V_\text{C}\$, then the gain is \$A_V\approx \frac{R_\text{C}}{R_\text{E}}\$.

But from the textbook's statement that the quiescent value of \$V_\text{E}=250\:\text{mV}\$ and assuming that \$I_\text{C}\approx I_\text{E}\$, we know the value of \$\frac{R_\text{E}}{R_\text{C}}=\frac{250\:\text{mV}}{5\:\text{V}}=0.05\$.

So using equation \$\ref{eq2}\$ to estimate the distortion, using the same peak values of \$V_\text{C}\$ as before, we get \$A_V\approx 18.12\$ @ \$V_\text{C}=5\:\text{V}\$, \$A_V\approx 18.08\$ @ \$V_\text{C}=5.1\:\text{V}\$, and \$A_V\approx 18.15\$ @ \$V_\text{C}=4.9\:\text{V}\$. Roughly, this is about a gain change of \$\pm 0.034\$ riding on an average gain of about \$18.1\$. That works out to about \$\pm 0.2\:\%\$. Dividing this value by 3, you get a figure of about \$0.07\:\%\$. Which is very close to the textbook's statement about the results of the analyzer's figure.

If you redo the calculations now using \$\pm 1\:\text{V}\$ variation rather than \$\pm 100\:\text{mV}\$ variation, you'll again find that the results are quite similar to their analyzer results.