Question

the box-and-whisker plot below represents some data set. what percentage of the data values are less than or equal to 110?

(box-and-whisker plot image with number line from 0 to 200, marked at 50, 100, 150)

answer attempt 1 out of 10

Explanation:

Step1: Recall box - and - whisker plot components

A box - and - whisker plot divides data into four quartiles, each representing 25% of the data. The median (second quartile, \(Q_2\)) divides the data into lower and upper halves. The first quartile (\(Q_1\)) is the median of the lower half, and the third quartile (\(Q_3\)) is the median of the upper half.
Looking at the box - and - whisker plot, we assume that 110 is at the third quartile (\(Q_3\)) or beyond? Wait, no, let's analyze the plot. Wait, the key property of a box - and - whisker plot is that:

• The minimum to \(Q_1\): 25% of data.
• \(Q_1\) to \(Q_2\): 25% of data (total 50% up to \(Q_2\)).
• \(Q_2\) to \(Q_3\): 25% of data (total 75% up to \(Q_3\)).
• \(Q_3\) to maximum: 25% of data.

Wait, if 110 is at or above \(Q_3\)? Wait, no, let's look at the number line. Wait, the box is split? Wait, maybe the 110 is at the \(Q_3\) or maybe the plot is such that the median is between the two boxes? Wait, no, the standard box - and - whisker plot has a box with \(Q_1\), \(Q_2\) (median), and \(Q_3\). Wait, maybe the 110 is at the \(Q_3\) position? Wait, no, let's think again.

Wait, actually, in a box - and - whisker plot, the data is divided into four equal parts (quartiles), each 25%. So:

• 25% of data is below \(Q_1\).
• 25% of data is between \(Q_1\) and \(Q_2\) (total 50% below \(Q_2\)).
• 25% of data is between \(Q_2\) and \(Q_3\) (total 75% below \(Q_3\)).
• 25% of data is between \(Q_3\) and maximum.

Wait, if 110 is at \(Q_3\), then the percentage of data less than or equal to 110 is 75%? Wait, no, maybe I made a mistake. Wait, let's check the plot again. The number line has 0, 50, 100, 150, 200. The box is around, maybe the \(Q_3\) is at 110? Wait, no, maybe the key is that in a box - and - whisker plot, the third quartile (\(Q_3\)) represents the value where 75% of the data is less than or equal to it. Wait, let's confirm:

The quartiles divide the data as follows:

• \(Q_1\): 25th percentile (25% of data ≤ \(Q_1\))
• \(Q_2\): 50th percentile (50% of data ≤ \(Q_2\))
• \(Q_3\): 75th percentile (75% of data ≤ \(Q_3\))

So if 110 is at \(Q_3\), then the percentage of data less than or equal to 110 is 75%. Wait, but maybe the plot is such that 110 is at \(Q_3\). Let's assume that the 110 is the third quartile (\(Q_3\)). Then, by the definition of quartiles, 75% of the data is less than or equal to \(Q_3\).

Step2: Determine the percentage

Since in a box - and - whisker plot, the third quartile (\(Q_3\)) marks the point where 75% of the data is less than or equal to it (because the data is divided into four quartiles, each 25%, so \(Q_1\) (25%), \(Q_2\) (50%), \(Q_3\) (75%)). If 110 is at \(Q_3\), then the percentage of data values less than or equal to 110 is 75%.

Answer:

75%