Measures of Dispersion for Ungrouped Data · 8.2.3
Box plots for ungrouped data
Students find the five-number summary, minimum, first quartile, median, third quartile and maximum, of an ungrouped data set, then draw it as a box-and-whisker diagram on a suitable scale. They must also read and interpret the finished box plot, describing how the data is spread and whether the distribution leans toward higher or lower values.
The official learning standard (8.2.3)
“Construct and interpret the box plot for a set of ungrouped data.”
What it means
Students find the five-number summary, minimum, first quartile, median, third quartile and maximum, of an ungrouped data set, then draw it as a box-and-whisker diagram on a suitable scale. They must also read and interpret the finished box plot, describing how the data is spread and whether the distribution leans toward higher or lower values.
How it is examined
In Paper 2, students are usually given a set of raw ungrouped data, asked to calculate the five-number summary, construct the box plot on graph paper, and then interpret it, for example, describing skewness or comparing spread with another data set. Paper 1 may ask students to read specific values directly off a given box plot.
Worked example
The marks of 9 students in a quiz are 42, 55, 58, 60, 63, 65, 70, 74 and 90. Determine the five-number summary, then state whether the distribution is skewed.
- Arrange the data in ascending order: 42, 55, 58, 60, 63, 65, 70, 74, 90 (already ordered).
- Minimum = 42, Maximum = 90.
- Median = 5th value (middle of 9 values) = 63.
- Q1 = median of the lower 4 values (42, 55, 58, 60) = (55 + 58) ÷ 2 = 56.5.
- Q3 = median of the upper 4 values (65, 70, 74, 90) = (70 + 74) ÷ 2 = 72.
- Compare the two halves of the box: median − Q1 = 63 − 56.5 = 6.5, while Q3 − median = 72 − 63 = 9, and the upper whisker (72 to 90 = 18) is longer than the lower whisker (42 to 56.5 = 14.5).
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I find Q1 and Q3 if there is an odd number of data values?
Exclude the median, then split the remaining data into a lower half and an upper half. Q1 is the median of the lower half, and Q3 is the median of the upper half.
If either half has an even count, average its two middle values.
What does a longer whisker or box on one side tell me?
It shows the data is more spread out on that side. A longer upper whisker or box means the distribution is skewed to the right (positively skewed); a longer lower side means it is skewed to the left (negatively skewed).
Do I need graph paper to draw a box plot in the exam?
Yes, draw it on a clear horizontal scale with the box plot aligned to your five-number summary. Label the minimum, Q1, median, Q3 and maximum clearly, and keep the scale even so distances on your diagram reflect the actual values.