What's the proper way for calculating first quartile? How should I calculate quartiles? In a book I read (by Amir D. Aczel) it is said that: > First quartile is th 25th percentile, that means a value for which $1/4$ of oberervation's results are lower. Shouldn't it be `lower or equal`? According to the book if I have observation like this: $1, 3, 7, 9, 10, 11$ the value of the first quartile is on position: $(6+1)\cdot 0.25 = 1.75$, so it's $1.75$. But shouldn't it be $3$? For both $1.75$ and $3$ 25% of observations are below the given value. But that's the same for $1.25, 2.9$ and $2$. Are all answers correct?

In theory it means the point a quarter of the way up the cumulative distribution. In practice there are various definitions used, particularly in statistical computing. For example in R there are nine different definitions, the first three for a discrete interpretation and the rest for a variety of continuous interpolations. Wikipedia has a tenth.

R types 1, 2, 3 and 5 give $3$ as the first quartile, as you calculated, and if you want to regard this observation as a discrete population that is indeed the obvious answer. The other interpolations range from $2$ to $4$.

Your figure of $1.75$ looks wrong and it is unclear how you calculated it: if it supposed to be a weighted average of $1$ and $3$ then perhaps you were trying to get $1 \times \frac14 + 3 \times \frac34 = 2.5$, which is what R type 6 would produce. Or perhaps you were trying something else.