Interquartile-Range Outliers

Interquartile-Range Outliers

English translation by SOTA – AI Community of the Russian original. Organisers who would like this translation removed can email [email protected].

Task 4 of the school stage of the All-Russian School Olympiad (VsOSh) 2025/26 in artificial intelligence (region group II), grades 9–11. The official answer and solution are in a separate file.

An array of $n$ numbers is given. Let us arrange them in ascending order:

$$x_1 \le x_2 \le x_3 \ldots \le x_n$$

The median is the middle of the ordered set: for odd $n$ it is the middle element; for even $n$ it is the mean of the two middle elements.
The first quartile $Q_1$ is the median of the lower half.
The third quartile $Q_3$ is the median of the upper half.
An array of $n$ numbers is given. Let us arrange them in ascending order:

$$x_1 \le x_2 \le \cdots \le x_n.$$

If $n$ is even, the lower half is $x_1, \ldots, x_{(n/2)}$ and the upper half is $x_{(n/2+1)}, \ldots, x_{(n)}$. If $n$ is odd, the element $x_{((n+1)/2)}$ does not belong to either half: the lower half is $x_1, \ldots, x_{((n-1)/2))}$ and the upper half is $x_{((n+3)/2)}, \ldots, x_{(n)}$. In all cases, if the median falls between numbers, the mean of the neighbouring ones is taken. The interquartile range is the length of the segment between the quartiles:

$$R = Q_3 - Q_1.$$

For a parameter $k > 0$, we define the bounds:

$$L = Q_1 - kR, \qquad U = Q_3 + kR.$$

A number is considered an outlier if it is less than or equal to $L$ or greater than or equal to $U$. This is what an example looks like for a different dataset.

[Figure: see the original statement.]

Translator's transcription of the figure: a number line from −1 to 22 with data points (crosses) at 0, 1, 2, …, 12 and (blue squares) at 16, 20 and 21. $Q_1 = 3.5$ and $Q_3 = 11.5$ are marked with diamonds, and the shaded band between them is labelled “Interquartile range”. Dotted lines mark $L$ (at −0.5) and $U$ (at 15.5). The points 16, 20 and 21 to the right of $U$ are braced together and labelled “outliers”.

Consider the array of 12 numbers:

$$\{12,\ -5,\ 14,\ 8,\ 30,\ 13,\ 9,\ 12,\ 11,\ 9,\ 12,\ 13\}.$$

It is known that the values of the parameter $k > 0$ for which exactly two elements of the array are outliers under the rule above form a half-open interval.

Write down the value of the left end of this half-open interval. Write down the value of the right end of the half-open interval.

Scoring criterion: 6 points for each correct answer. Total — 12 points

Maximum score for the task — 12