36 exam-style questions on Data presentation: histograms, box plots, cumulative frequency — topic 4.2 of Statistics and Probability in the IB Mathematics: Analysis and Approaches syllabus — with answers and explanations. Samples below; the full set is free with an account.
A box-and-whisker plot has minimum 4, lower quartile 10, median 14, upper quartile 20 and maximum 30. Calculate the interquartile range.
Answer: 10
IQR = Q3 − Q1 = 20 − 10 = 10; the range (26) and Q3 − median (6) are common mix-ups.
A cumulative frequency graph shows the masses of 80 parcels. State the cumulative frequency value at which the upper quartile should be read.
Answer: 60
Q3 corresponds to 3/4 of the total frequency: 3/4 × 80 = 60; reading at 40 would give the median instead.
A data set has lower quartile 20 and upper quartile 32. Using the criterion that an outlier lies more than 1.5 × IQR beyond the nearer quartile, determine which of the following values is an outlier.
Answer: 52
IQR = 12, so the upper boundary is 32 + 1.5 × 12 = 50; only 52 lies beyond this boundary (50 itself is not an outlier).
On a cumulative frequency graph of journey times, the cumulative frequency is 26 at 20 minutes and 68 at 40 minutes. Determine how many journeys took between 20 and 40 minutes.
Answer: 42
The number of values in an interval is the difference of the cumulative frequencies: 68 − 26 = 42; adding them (94) is a classic error.
In a frequency histogram with equal class widths, state what the height of each bar represents.
Answer: The frequency of the class
With equal class widths the bar height is simply the class frequency; cumulative frequency is shown on a cumulative frequency curve, not a histogram.
In a box-and-whisker plot, the median is much closer to the lower quartile than to the upper quartile, and the right-hand whisker is longer than the left. Suggest the most appropriate description of the distribution.
Answer: Positively skewed
A longer tail towards higher values, with the median pulled towards Q1, indicates positive skew; a box plot cannot show whether data are bimodal.
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