Interquartile range

As the varied answers indicate, extracting quantiles is something of an inexact science when you have few enough samples that the rounding between two neighbor samples matters.

A bit abstractly expressed, you have tabulated values for $f(1)$, $f(2)$, ... $f(18)$, but getting from there to actual quartiles requires at least two semi-arbitrary choices:

1. How do we define values of $f$ for non-integral arguments when "a quarter way through the sample set" happens not to hit one particular sample exactly? Linear interpolation between neighbor samples is a popular choice, but it seems that Wolfram Alpha instead extends $f$ to a step function. Even step functions can be done in different ways: round up? round down? round to nearest? In the latter case, what about the point exactly halfway between samples?

2. What is actually the interval that we want to find quarter-way points in? One natural choice is $[1,18]$, which makes the zeroth and fourth quartile exactly the minimum and maximum. But a different natural choice is $[0.5, 18.5]$ such that each sample counts for the same amount of x-axis. In the latter case there is a risk that one will have to find $f(x)$ for $x<1$ or $x>18$, where a linear interpolation does not make sense. More decisions to make then.

It looks like your book is using yet a third interval, namely $[0, 19]$! Then, by linear interpolation, we get $$Q1 = f(4.75) = 5+0.75\times(8-5) = 7.25$$ $$Q3 = f(14.25) = 23+0.25\times(25-23) = 23.5$$

I'm not sure how you get your own suggestions for quartiles. Since you divide 18 by 4, I assume you use an interval of length 18, but if you're using linear interpolation, you compute Q1 as $f(4.5)$ and Q2 as $f(9.5)$, with a distance of only 4 rather than 4.5. Or are you completing $f$ such that every non-integral $x$ maps to the midpoint between neighbor samples?