Probability problem with box chart I am tackling a problem of box chart as shown below:



*

*IB grades can only take Internet values


Here are the questions:

(A) How many students obtained a grade of more than $4$?

For this question, I thought the left bottom of the box represent $25\%$ of the students, therefore the students that got a grade above $4$ is equal to $75\%.$ Is that correct?

(B) State, with reasons, the maximum possible number and minimum possible number of students who obtained a $4$ in the exam.

How can I solve this problem? I have no ideas! Help!
 A: For the first question, you're correct. The left edge of the box represents the first quartile, $Q_1$, which is the $25^{th}$ percentile.

To reason your way through the second question, think about what you know about what values were scored.


*

*You know there are $100$ test scores

*At least one was a $2$, and it was the lowest

*At least one was a $4$, which is the first quartile

*At least one was a $6$, which is the third quartile

*At least one was a $7$, which is the peak

*The median of the dats is $4.5$
The quartiles themselves are dependent on order - the first quartile (roughly) is the median of the data less than the median, and the third is (roughly) the median of the data greater than the median. In this case, the $25^{th}$ and $75^{th}$ values once the test scores were ordered.
So we know the score of $4$ is the first quartile. Thus, the $25^{th}$ value is a $4$. We know the lowest is a $2$, we don't know about $3$'s, and we know that the median is $4.5$.
From the latter fact, we can conclude that the $50^{th}$ and $51^{st}$ data points are $4$ and $5$ respectively, or $3$ and $6$, or $2$ and $7$, since we can't have non-integer* values, and the number of scores is even so we'll be taking an average. We can safely eliminate the latter two pairs since $Q_1 = 4$.
($*$ - I assume IB scores are integer-only values, anyhow. I'm not sure myself having never dealt with it. I addressed this in the comments of the OP but hadn't gotten a response by the time I wrote this up. If my assumptions is wrong, I imagine the overall "idea" of this post holds, even if the actual math no longer does.)
So we know that the $50^{th}$ and $25^{th}$ values are $4$ in the ordered data, and thus all those data points in-between are definitely $4$. We know, beyond the $50^{th}$ data point, we have $5$ or higher.
Can we say anything about the data lower than $Q_1$? We know the lowest data point was a $2$, but nothing else. So we can make the data points as extreme as we want without affecting the plot, to an extent, so let's see.
Hypothetically, data points $\#1-\#24$ could all be $2$ (no $4$'s whatsoever below $Q_1$). This would mean that we have $26$ $4$-point tests - tests $\#25-\#50$.
Hypothetically, we could have $\#1$ be $2$, but then $\#2-\#24$ all be $4$ (we'd get all $4$'s below $Q_1$ until the minimum). this would add a further $23$ tests, thus yielding $49$ $4$-point tests.
Thus, the possible range of frequencies of $4$ is $26$ to $49$.
