Question

question
the box-and-whisker plot below represents some data set. what percentage of the data values are between 8 and 18?
answer attempt 1 out of 2
submit answer

Explanation:

Step1: Recall Box - Whisker Plot Basics

In a box - whisker plot, the whiskers represent the minimum and maximum values, and the box has the first quartile ($Q_1$), median ($Q_2$), and third quartile ($Q_3$). The data is divided into four parts (quartiles), each representing 25% of the data. The left whisker goes from the minimum to $Q_1$, the box from $Q_1$ to $Q_3$ (with median inside), and the right whisker from $Q_3$ to maximum. Wait, actually, the left whisker is from min to $Q_1$ (25% of data below $Q_1$), the box is from $Q_1$ to $Q_3$ (50% of data between $Q_1$ and $Q_3$), and the right whisker is from $Q_3$ to max (25% of data above $Q_3$). Wait, no, correction: The minimum to $Q_1$: 25% of data (since quartiles divide data into four equal parts). $Q_1$ to median: 25%, median to $Q_3$: 25%, $Q_3$ to maximum: 25%. Wait, actually, the box contains the middle 50% of the data (from $Q_1$ to $Q_3$), and each whisker (left and right) contains 25% of the data (left whisker: min to $Q_1$, 25% of data; right whisker: $Q_3$ to max, 25% of data). Wait, no, the correct breakdown is: The entire data set is divided into four quartiles. The first quartile ($Q_1$) is the value where 25% of the data is below it, the median ($Q_2$) is where 50% of the data is below it, and the third quartile ($Q_3$) is where 75% of the data is below it. So the data between min and $Q_1$: 25% (since 25% of data is below $Q_1$), between $Q_1$ and $Q_2$: 25%, between $Q_2$ and $Q_3$: 25%, between $Q_3$ and max: 25%.

Looking at the plot, the left whisker starts at 8 (min) and goes up to $Q_1$ (let's assume the left end of the box is $Q_1$). Wait, the left whisker's top (the start of the box) is at, looking at the graph, the left side of the box is at, say, 8? Wait, no, the left whisker is from min (let's say min is 8) to $Q_1$ (the start of the box). Wait, the problem is about data between 8 and 18. Let's assume that 8 is the minimum (start of left whisker) and 18 is $Q_1$ (start of the box). Then the data from min (8) to $Q_1$ (18) would be 25% of the data? Wait, no, if 8 is the minimum and 18 is $Q_1$, then the data between 8 (min) and 18 ($Q_1$) is the first quartile, which is 25% of the data. Wait, but maybe I misread. Wait, the left whisker: from min to $Q_1$ (25% of data), box from $Q_1$ to $Q_3$ (50% of data), right whisker from $Q_3$ to max (25% of data). So if 8 is the min and 18 is $Q_1$, then the data between 8 and 18 is the first quartile, which is 25% of the data? Wait, no, quartiles: 25% of data is less than or equal to $Q_1$, 50% less than or equal to median, 75% less than or equal to $Q_3$, 100% less than or equal to max. So the proportion of data between min and $Q_1$ is 25% (since $Q_1$ is the 25th percentile). So if 8 is the min and 18 is $Q_1$, then the data between 8 and 18 is 25% of the data. Wait, but maybe the left whisker is from min (let's say 8) to $Q_1$ (18), so the data in that interval is 25%? Wait, no, maybe I made a mistake. Wait, let's re - express:

In a box - and - whisker plot:

- The minimum value (start of left whisker) to $Q_1$: 25% of the data (because $Q_1$ is the 25th percentile, meaning 25% of data is at or below $Q_1$).

- $Q_1$ to median: 25% of the data (since median is 50th percentile, so from 25th to 50th percentile is 25% of data).

- Median to $Q_3$: 25% of the data (from 50th to 75th percentile).

- $Q_3$ to maximum (end of right whisker): 25% of the data (from 75th to 100th percentile).

Now, if 8 is the minimum (start of left whisker) and 18 is $Q_1$ (start of the box), then the data between 8 and 18 is the data from min to $Q_1$, which is 25% of the data. Wait, but maybe the left whisker is from min (8) to $Q_1$ (18), so the percentage of data between 8 and 18 is 25%? Wait, no, maybe I got the quartiles reversed. Wait, no, the left whisker is the lower 25% (from min to $Q_1$), the box is the middle 50% (from $Q_1$ to $Q_3$), and the right whisker is the upper 25% (from $Q_3$ to max). So if 8 is the min and 18 is $Q_1$, then the data between 8 and 18 is 25% of the data. But wait, maybe the question is that 8 is the min and 18 is within the left whisker or the box? Wait, the graph: the left whisker has a vertical line at the top (the start of the box) at, looking at the y - axis (the number line), the left end of the box is at, say, 8? No, the left whisker is from, like, 5 (but the problem says 8) to the start of the box. Wait, the problem says "between 8 and 18". Let's assume that 8 is the minimum (start of left whisker) and 18 is the first quartile ($Q_1$, start of the box). Then the data from 8 (min) to 18 ($Q_1$) is 25% of the data, because quartiles divide the data into four equal parts (25% each). So the percentage of data between 8 and 18 is 25%? Wait, no, maybe I messed up. Wait, another way: In a box - whisker plot, the left whisker represents the data from the minimum to the first quartile ($Q_1$), which is 25% of the data. The box represents the data from $Q_1$ to the third quartile ($Q_3$), which is 50% of the data. The right whisker represents the data from $Q_3$ to the maximum, which is 25% of the data. So if 8 is the minimum and 18 is $Q_1$, then the data between 8 and 18 is the data in the left whisker, which is 25% of the data. But wait, maybe 18 is inside the box? No, the problem is about between 8 and 18. Let's check the number line: the left whisker's top (start of box) is at, say, 8? No, the left whisker has a vertical line (the whisker) from, like, 5 to 10? Wait, the image shows the left whisker's top (the start of the box) is at around 8? Wait, the y - axis has marks at 0, 10, 20, 30, 40. The left whisker is a vertical line (the whisker) from, say, 5 to 10? No, the box starts at, looking at the graph, the left side of the box is at, maybe, 8? Wait, the problem says "between 8 and 18". Let's assume that 8 is the minimum (the bottom of the left whisker) and 18 is the first quartile (the top of the left whisker, start of the box). Then the data from 8 to 18 is the left whisker, which is 25% of the data. So the percentage is 25%? Wait, no, maybe I made a mistake. Wait, maybe 8 is the minimum and 18 is within the left whisker, but no, quartiles: 25% below $Q_1$, 25% between $Q_1$ and median, 25% between median and $Q_3$, 25% above $Q_3$. So if 8 is min and 18 is $Q_1$, then 25% of data is between 8 and 18. So the answer should be 25%? Wait, but maybe I got the direction wrong. Wait, no, let's think again. The key is that in a box - whisker plot, each quartile (the space between min - $Q_1$, $Q_1$ - median, median - $Q_3$, $Q_3$ - max) has 25% of the data. So if 8 is the min and 18 is $Q_1$, then the data between 8 and 18 is the first quartile, which is 25% of the data. So the percentage is 25%.

Step2: Confirm with Quartile Definition

Quartiles divide a data set into four equal parts, each containing 25% of the data. The left - most part (from minimum to first quartile, $Q_1$) contains 25% of the data. If 8 is the minimum value (start of the left whisker) and 18 is the first quartile (end of the left whisker, start of the box), then the data values between 8 and 18 lie within this first 25% of the data. So the percentage of data values between 8 and 18 is 25%.

Answer:

25%