The site correctly shows an aspect ratio of 16:10 for the X250, which is unusual compared to the far more common 16:9. However, it doesn't show any aspect ratio for the X260, so it probably assumes the far more common 16:9 as a default value.

A 12.5" diagonal with an aspect ratio of 16:9 means a screen area of 66.8 in^2. So, the square-pixels-per-square-inch is (1920 pixels * 1080 pixels) / (66.8 in^2). Take the square root of that to get pixels-per-inch (PPI), and it's 176, just like the website shows.

The math for the X260's PPI looks correct given that it's missing the aspect ratio and probably just defaulting to the common 16:9.

PS - Wikipedia's equation for PPI, as cited in the currently accepted answer, is simply wrong. PPI is the square root of the number of pixels over the area in which they're displayed. This doesn't reduce to the expression given on Wikipedia.