Creating and Interpreting Box Plots - Sixth Grade Mathematics
A box-and-whisker plot (box plot) is a visual representation that displays the distribution and spread of a data set along a number line using the five-number summary.
The Anatomy of a Box Plot
A box plot divides data into four distinct quartiles, with each section representing exactly 25% of the total observations regardless of the physical length of the segment.
Anatomy of a Box-and-Whisker Plot:
|-------------- Middle 50% (IQR) -------------|
| |
| +---------------+ |
*---------+-------------| | |---------------+---------*
| +---------------+ |
Min Q1 Median Q3 Max
(25%) (Q2/50%) (75%)
|<----- 25% ----------->|< 25% >|< 25% >|<--------- 25% --------->|
Lower Whisker Lower Upper Upper Whisker
Box Box
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+---->
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Key Components:
1. The Box: Stretches from Q1 to Q3, enclosing the middle 50% of the data.
Its total width represents the Interquartile Range (IQR = Q3 - Q1).
2. The Median Line: A vertical line drawn inside the box at the exact location of Q2.
3. The Whiskers: Horizontal line segments extending outward from the box to the
minimum and maximum values (representing the lower 25% and upper 25%).
Constructing a Box Plot Step-by-Step
To construct an accurate box plot: Step 1: Sort raw data in ascending order. Step 2: Calculate the Five-Number Summary: Min, Q1, Median, Q3, Max. Step 3: Draw a continuous numerical scale (number line) with equal intervals that spans from below the minimum to above the maximum. Step 4: Draw short vertical line segments above the number line at Q1, Median, and Q3, and connect them into a rectangular box. Step 5: Place dots or tick marks at the Minimum and Maximum values, and draw horizontal whiskers connecting them to the sides of the box.
Worked Construction Example:
Data (Quiz scores): 14, 16, 18, 20, 22, 24, 26, 28, 30
1. Min = 14
2. Lower Half: 14, 16, 18, 20 --> Q1 = (16 + 18)/2 = 17
3. Overall Median = 22
4. Upper Half: 24, 26, 28, 30 --> Q3 = (26 + 28)/2 = 27
5. Max = 30
Five-Number Summary: 14, 17, 22, 27, 30.
IQR = 27 - 17 = 10. Range = 30 - 14 = 16.
Interpreting Quartile Percentages and Density
A common misconception is thinking that a longer whisker or wider box section contains more data points. Every single section of a box plot contains exactly 25% of the data!
Density vs. Spread:
- A SHORT section means 25% of the data points are packed tightly together within a narrow range of numbers (high density).
- A LONG section means 25% of the data points are spread out widely across a broad range of numbers (low density).
If the median is located closer to Q1, the lower middle data is clustered and the distribution is skewed right. If the median is centered between Q1 and Q3, the distribution is symmetric.
Parallel Box Plots for Comparison
Plotting two box plots against the identical number line allows direct visual comparison between two distributions.
Comparing Heights: Grade 6 vs. Grade 8:
Grade 6: *-------[ | ]-------*
50 54 57 60 64
Grade 8: *-------[ | ]-------*
58 62 65 68 72
----+----+----+----+----+----+----+----+----+----+----+----+----+---->
48 50 52 54 56 58 60 62 64 66 68 70 72 Inches
Visual comparison shows: The median height of Grade 8 (65 in.) is higher than the entire upper quartile of Grade 6. Approximately 75% of Grade 8 students are taller than the median Grade 6 student.
Exercises
Exercise 1. A box plot has a minimum of 10, Q1 = 18, Median = 25, Q3 = 34, and maximum = 45. (a) What percent of data lies between 18 and 34? (b) What percent of data is greater than 25? (c) What is the IQR? (d) What is the overall range?
Exercise 2. Construct a five-number summary and determine the IQR for the data set: 12, 15, 16, 19, 21, 24, 27, 30, 32, 35.
Exercise 3. Explain why a whisker that is twice as long as another whisker does not mean it has twice as many data points.
Solutions
Solution to Exercise 1. For (a), the region between Q1 (18) and Q3 (34) represents the central box, which contains exactly 50% of the data. For (b), the median (25) splits the distribution in half, so 50% of the data is greater than 25. For (c), IQR = Q3 - Q1 = 34 - 18 = 16. For (d), Range = Max - Min = 45 - 10 = 35.
Solution to Exercise 2. Count = 10 (even). Min = 12, Max = 35. Median = (21 + 24)/2 = 22.5. Lower half: 12, 15, [16], 19, 21 -> Q1 = 16. Upper half: 24, 27, [30], 32, 35 -> Q3 = 30. Five-Number Summary: Min = 12, Q1 = 16, Median = 22.5, Q3 = 30, Max = 35. IQR = Q3 - Q1 = 30 - 16 = 14.
Solution to Exercise 3. Every quartile segment in a box plot—whether it is a whisker or half of the central box—always contains exactly one quarter (25%) of the total data points. A longer whisker simply indicates that those 25% of data values are spread out over a wider numerical distance, meaning the data in that region is less densely packed.