Two classes took the same test. Class A's box plot has a median of 75, Q1 = 65, Q3 = 85, with a long whisker extending to 40 on the left. Class B's box plot has a median of 78, Q1 = 72, Q3 = 84. What can you conclude?
AClass A has more students who scored below 65 than Class B does
BClass A's distribution is more spread out in the lower half than Class B's, but about 25% of Class A scored between 40 and 65
CThe long left whisker in Class A means more students scored low than in Class B
DClass B performed better because its median is higher
The long left whisker of Class A indicates spread, not count. Approximately 25% of Class A's students scored between the minimum (40) and Q1 (65) — that's the same proportion as any other quartile section. The whisker is long because those scores are spread across a wide range (40 to 65), not because more students scored there. Class B's narrow IQR (72 to 84) means the middle 50% are tightly clustered. The medians are close (75 vs 78), so option D oversimplifies.
Question 2 Multiple Choice
A data set has Q1 = 50 and Q3 = 80. What does the IQR of 30 tell you?
AThe range of the entire data set is 30
BThe middle 50% of the data values fall within a 30-unit span
CThe average value in the data set is around 65
DThere are 30 data values between Q1 and Q3
The IQR (Q3 − Q1 = 80 − 50 = 30) measures the spread of the middle 50% of the data — it is the width of the box. It tells you nothing about the number of data points in that range (each quartile always contains about 25% of the data regardless of IQR width) and nothing about the mean (option C) or total range (option A). A wide IQR means the middle half is spread out; a narrow IQR means it is tightly clustered.
Question 3 True / False
In a box-and-whisker plot, a longer whisker on one side means there are more data points in that region than in a shorter whisker region.
TTrue
FFalse
Answer: False
This is the most common misconception about box plots. Each section of a box plot — including each whisker — contains approximately 25% of the data values, regardless of its length. A long whisker means the data in that quartile is spread across a wider range of values; a short whisker means those same 25% of values are clustered close together. Length signals spread, not frequency.
Question 4 True / False
Each of the four sections of a box plot (min to Q1, Q1 to median, median to Q3, Q3 to max) contains approximately 25% of the data values, regardless of how wide each section appears.
TTrue
FFalse
Answer: True
This is the foundational principle that makes box plots useful: quartiles divide the data into four equal-frequency groups. By definition, Q1 is the value below which 25% of data falls, Q2 (median) is the midpoint, and Q3 is the value below which 75% falls. So each of the four sections contains about 25% of values. The visual widths of those sections reflect how spread out the values are, not how many values are in each section.
Question 5 Short Answer
A box plot shows the median line very close to Q3 (the right side of the box), with Q1 much farther to the left. What does this reveal about the shape of the distribution?
Think about your answer, then reveal below.
Model answer: When the median sits close to Q3, the middle half of the data is clustered near the upper end of the box. This means the data is left-skewed (negatively skewed): there is a long tail on the lower end, with most values concentrated toward the higher end of the range. The asymmetric position of the median within the box is the key visual signal — a centered median suggests a symmetric distribution, while a median shifted toward one quartile suggests skewness in the opposite direction.
Reading skewness from a box plot is one of its most practical applications. A histogram shows shape more clearly for a single data set, but a box plot enables fast comparison of skewness across multiple groups on the same scale. If Class A has a median near Q3 and Class B has a centered median, you immediately know Class A's scores are more concentrated at the high end while Class B's are more symmetric — insights that a simple comparison of means would miss.