Question 3·Hard·One-Variable Data Distributions; Measures of Center and Spread
The box plots summarize the completion times, in minutes, for two different classes taking the same exam.
Based on the box plots, which of the following statements must be true?
For box-plot questions, first recall what each part represents: the median is the line inside the box, the box length is the interquartile range, and the whisker-to-whisker length is the range; the mean is not shown at all. For a 'must be true' question, immediately eliminate any choice about a statistic that is not directly visible (like the mean), then visually compare medians, box lengths, and total lengths to see which inequality (greater/less) actually matches the graph. This targeted matching-and-elimination approach saves time and reduces errors from overthinking unseen details of the data. Роwеrеd bу Аnікo
Hints
Connect answer choices to features of a box plot
Identify which parts of the box plot show each of these: mean, median, interquartile range, and range. Ask yourself: which of these is actually drawn on the box plot?
Use the idea of “must be true”
A statement that must be true has to hold for any data set that would produce these box plots. If the graph does not directly show a value (like the mean), can you be certain about how it compares between the two classes? Frοm аniкo.аі
Look at center, middle spread, and overall spread
Compare three things on the two box plots: the position of the median lines (center), the lengths of the boxes (middle 50% of the data), and the total whisker-to-whisker lengths (overall spread). Which comparison clearly matches one of the statements?
Desmos Guide
Record quartile values from the box plots
From the box plots, read off approximate numerical values for each class’s first and third quartiles: Q1 and Q3 for Class X, and Q1 and Q3 for Class Y. Write them down. Sоurсe: anіко.аі
Enter quartiles into Desmos
In Desmos, define the quartiles as variables, for example:
- Type
Q1x =followed by your Q1 value for Class X. - Type
Q3x =followed by your Q3 value for Class X. - Type
Q1y =followed by your Q1 value for Class Y. - Type
Q3y =followed by your Q3 value for Class Y.
Compute and compare interquartile ranges
In Desmos, type IQRx = Q3x - Q1x and IQRy = Q3y - Q1y. Compare the two results: decide which class has the larger IQR, then choose the answer choice that matches that relationship between the classes.
Step-by-step Explanation
Recall what information a box plot shows
A box plot summarizes a data set using 5 key numbers:
- minimum (left end of the left whisker)
- first quartile, Q1 (left edge of the box)
- median (line inside the box)
- third quartile, Q3 (right edge of the box)
- maximum (right end of the right whisker)
The box runs from Q1 to Q3, so its length represents the interquartile range (IQR), with . The full distance from the minimum to the maximum (from the left whisker end to the right whisker end) represents the range.
Match each answer choice to what it is asking about
Go through the choices and link each one to the part of the box plot it refers to:
- Choice A talks about the mean. The mean is not displayed anywhere on a standard box plot.
- Choice B talks about the median, which is the vertical line inside each box.
- Choice C talks about the interquartile range, which is the length of the box from Q1 to Q3.
- Choice D talks about the range, which is the total distance from the minimum to the maximum (end of left whisker to end of right whisker).
Eliminate choices that cannot be guaranteed from box plots
Think about what is and is not determined by a box plot.
- Because the mean is not shown on a box plot, different data sets can have the same box plot but different means. That means choice A cannot be something that must be true.
- The box plot does fix the median, quartiles, and the minimum and maximum. So any statement about those can be checked directly against the graph, but it still has to match what the plot actually shows (greater, less, or equal).
Compare medians, ranges, and box lengths to select the correct statement
Now use the actual plots to check the remaining types of comparisons:
- Look at the median lines inside the boxes. In the plots, the median for Class X is not greater than the median for Class Y, so choice B is not correct.
- Look at the full whisker-to-whisker lengths. The overall spread for Class X is not less than that for Class Y, so choice D is not correct.
- Finally, compare the lengths of the boxes (from Q1 to Q3). The box for Class X is clearly longer than the box for Class Y, meaning Class X’s IQR is greater than Class Y’s IQR.
Therefore, the statement that must be true is: “The interquartile range of Class X is greater than the interquartile range of Class Y.”