Question

question the box-and-whisker plot below represents some data set. what percentage of the data values are between 40 and 55?

Explanation:

Step1: Recall box - whisker plot properties

In a box - and - whisker plot, the box represents the interquartile range (IQR), which contains the middle 50% of the data. The median (the line inside the box) divides the data into two halves, each containing 50% of the data. If we consider the data between the median (40, assuming the median line is at 40) and the upper quartile (Q3). Wait, actually, looking at the box - whisker plot: the median is at 40, and the upper part of the box (from median to Q3) or lower? Wait, no, let's re - analyze. The box has a line (median) at 40. The box itself spans from, let's see, the lower quartile (Q1) to upper quartile (Q3). But the question is about data between 40 (median) and 55? Wait, no, maybe the median is 40, and the upper half of the box? Wait, no, in a box - whisker plot, the median splits the data into two groups of 50% each. If we are looking at data between the median (40) and, say, the upper quartile? Wait, no, the key is that the median divides the data into two 50% parts. But if the data between 40 (median) and, let's assume that 40 is the median, and the data from median to the upper part: but actually, in a box - whisker plot, the box contains 50% of the data (from Q1 to Q3). The median is the middle of the box, so from median to Q3 is 25% of the data, and from Q1 to median is 25% of the data. Wait, no: the box is from Q1 (25th percentile) to Q3 (75th percentile), so the length of the box is IQR, and it contains 50% of the data. The median (50th percentile) is inside the box, splitting the box into two equal parts (each part represents 25% of the total data). So if the median is at 40, and we want data between 40 and, say, the upper part of the box (up to Q3), but in the plot, if the median is 40, and the data between 40 and 55: since the median is the 50th percentile, and the data above the median is 50%? No, wait, no. Wait, the total data is 100%. The median (50th percentile) splits the data into two groups: 50% below the median and 50% above the median. But if we are looking at data between the median (40) and a value within the upper half of the box (which is 25% of the total data, because the box is 50% and split by median into two 25% parts). Wait, no, let's correct:

- The box - whisker plot has: minimum, Q1 (25th percentile), median (50th percentile), Q3 (75th percentile), maximum.
- The data from Q1 to Q3 (the box) is 50% of the data (25th to 75th percentile).
- The median (50th percentile) is the middle of the box, so from Q1 to median: 25% of data (25th to 50th percentile), from median to Q3: 25% of data (50th to 75th percentile).

So if the median is at 40, and we want data between 40 (median) and, say, Q3 (which is within the range up to 55? Wait, the plot shows that the median is at 40, and the upper part of the box (from median to Q3) is 25% of the data. Wait, but maybe the question is that 40 is the median, and the data between 40 and 55 is the upper 25%? No, wait, no. Wait, the key is that the median divides the data into two equal parts (50% below, 50% above). But if the data between 40 (median) and, let's say, the upper quartile (Q3) is 25%? Wait, no, the box is Q1 to Q3 (50% of data), so median is in the middle of the box, so from median to Q3 is 25% of the total data (because 50% / 2 = 25%).

Wait, maybe the plot has the median at 40, and the data between 40 and 55 is the data from the median (50th percentile) to the upper quartile (75th percentile), which is 25% of the data? No, wait, no. Wait, the total data:

- Below Q1 (25th percentile): 25%
- Between Q1 and median (50th percentile): 25%
- Between median and Q3 (75th percentile): 25%
- Above Q3: 25%

So if we want data between 40 (median) and 55 (which is Q3, assuming), then the percentage is 25%? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, the median is the 50th percentile. So data above the median is 50% of the data. But if the upper quartile (Q3) is at 55, then data between median (40) and Q3 (55) is 25% (because from 50th to 75th percentile is 25% of the data). Wait, no, 75 - 50 = 25, so 25% of the data is between median (50th) and Q3 (75th).

Wait, let's re - express:

- Percentile 0: minimum
- Percentile 25: Q1
- Percentile 50: median
- Percentile 75: Q3
- Percentile 100: maximum

So the data between percentile 50 (median = 40) and percentile 75 (Q3) is 75 - 50 = 25% of the data.

Step2: Confirm with box - whisker plot structure

In a box - whisker plot, the box contains 50% of the data (from Q1 to Q3). The median line inside the box splits the box into two equal regions, each representing 25% of the total data (since 50% / 2 = 25%). So if the median is at 40, and we are looking at the data from the median (40) to the upper end of the box (Q3), this region contains 25% of the data.

Answer:

25%