Measures of Dispersion for Grouped Data · 7.2.2
Box plot for grouped data
Students find the five-number summary for grouped data, the lower boundary of the first class as the minimum, Q1, the median, Q3, and the upper boundary of the last class as the maximum, usually by interpolation or from an ogive, then draw a box plot to scale and interpret its skewness and spread.
The official learning standard (7.2.2)
“Construct and interpret a box plot for a set of grouped data.”
What it means
Students find the five-number summary for grouped data, the lower boundary of the first class as the minimum, Q1, the median, Q3, and the upper boundary of the last class as the maximum, usually by interpolation or from an ogive, then draw a box plot to scale and interpret its skewness and spread.
How it is examined
In Paper 2, students are typically given a grouped frequency table (or an already-drawn ogive), asked to find the five-number summary, construct the box plot on graph paper, and interpret it, describing skewness or comparing spread with another data set. Paper 1 may ask students to read a specific value directly off a drawn box plot.
Worked example
The table shows the marks of 40 students in a test: 1–10 (2), 11–20 (6), 21–30 (14), 31–40 (12), 41–50 (6). Find the five-number summary and construct a box plot, then comment on the shape of the distribution.
- N = 40. Cumulative frequencies: 2, 8, 22, 34, 40.
- Minimum = lower boundary of the first class = 0.5; maximum = upper boundary of the last class = 50.5 (grouped data does not give the true individual smallest and largest values).
- Q1 at N/4 = 10 falls in the class 21–30 (boundaries 20.5–30.5, cumulative frequency from 8 to 22): Q1 = 20.5 + [(10 − 8)/14] × 10 = 21.9.
- Median at N/2 = 20 also falls in the class 21–30: median = 20.5 + [(20 − 8)/14] × 10 = 29.1.
- Q3 at 3N/4 = 30 falls in the class 31–40 (boundaries 30.5–40.5, cumulative frequency from 22 to 34): Q3 = 30.5 + [(30 − 22)/12] × 10 = 37.2.
- Draw the box from Q1 to Q3 with a line at the median, and whiskers to the minimum and maximum: the left whisker (0.5 to 21.9) is noticeably longer than the right whisker (37.2 to 50.5), so the marks are skewed toward the lower end.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Where do the minimum and maximum come from if grouped data doesn't give individual values?
They are approximated using the lower boundary of the very first class as the minimum and the upper boundary of the very last class as the maximum. This is a convention that assumes the data could lie anywhere within its class, so a box plot for grouped data is always an estimate, not an exact picture.
How do I tell if a box plot is skewed just by looking at it?
Compare the two whiskers and the two halves of the box. If the whisker (or box half) on one side is clearly longer than the other, the distribution is skewed toward that longer side, a longer left whisker means low values pull the shape to the left (negative skew), and a longer right whisker means the opposite (positive skew).
Do I need to draw the ogive first before I can draw the box plot?
Not strictly, you can find Q1, the median and Q3 directly by interpolation from the cumulative frequency table without drawing anything. However, if a question already gives you an ogive, reading the quartiles off it is usually faster and is an accepted alternative to the interpolation formula.