# Where will a prediction interval or a confidence interval be narrower: near the mean or further from the mean?

$E = {t}_{\setminus \frac{\alpha}{2} , \mathrm{df} = n - 2} \setminus \times {s}_{e} \setminus \sqrt{\left(\setminus \frac{1}{n} + \setminus \frac{{\left({x}_{0} - \setminus \overline{x}\right)}^{2}}{{S}_{x x}}\right)}$
$E = {t}_{\setminus \frac{\alpha}{2} , \mathrm{df} = n - 2} \setminus \times {s}_{e} \setminus \sqrt{\left(1 + \setminus \frac{1}{n} + \setminus \frac{{\left({x}_{0} - \setminus \overline{x}\right)}^{2}}{{S}_{x x}}\right)}$
In both of these, we see the term ${\left({x}_{0} - \setminus \overline{x}\right)}^{2}$, which scales as the square of the distance of the prediction point from mean. This is why CI and PI is narrowest at the mean.