Year 10
Construct and interpret histograms, box plots and cumulative frequency diagrams fluently, and compare data sets using these representations with precision.
Statistics and data — distributions
A **histogram** shows the frequency density of grouped continuous data. The height represents frequency density = frequency ÷ class width. A **box plot** displays the quartiles and outliers, showing the median (Q₂), lower quartile (Q₁), upper quartile (Q₃), and any outliers. The interquartile range IQR = Q₃ − Q₁. A **cumulative frequency curve** plots the cumulative frequency (running total) against the upper class boundary, allowing us to read off the median and quartiles visually. These diagrams enable comparison of distributions: centre (median), spread (IQR), and shape.
Reading a frequency table and drawing a histogram
A histogram shows time taken (seconds), grouped into equal-width classes of 10: 0-10 (3), 10-20 (6), 20-30 (8), 30-40 (5), 40-50 (2). Class 20-30 has frequency 8, so its frequency density is 8 ÷ 10 = 0.8.
In a histogram, the height (frequency density) must account for the class width. With equal widths, the bars still look just like a bar chart -- the difference only becomes essential once widths vary.
Why unequal class widths need frequency density
Ages of 50 people at an event, grouped into unequal classes: 0-10 (6), 10-20 (14), 20-40 (20), 40-70 (8), 70-100 (2).
The 40-70 class has fewer people than 20-40, but if you plotted raw frequency as the height it would look almost as tall, since the class is wider. Dividing by width (frequency density) fixes this -- the bar's area, not its height, is what represents the frequency.
Drawing a box plot from data
Data: 2, 4, 5, 6, 7, 8, 9, 10, 12. Draw the box plot and identify Q₁, Q₂ (median), Q₃.
Find the quartiles by dividing the ordered data into quarters. The box spans Q₁ to Q₃, with a line for the median.
Reading from a cumulative frequency curve
100 test scores are grouped as: 0-20 (5), 20-40 (20), 40-50 (15), 50-60 (35), 60-70 (15), 70-100 (10). Estimate the median and IQR.
Cumulative frequency goes from 0 to n. The median is at n/2, Q₁ at n/4, Q₃ at 3n/4.
Two groups took a test. Group A: min 45, Q₁ 65, median 72, Q₃ 80, max 92. Group B: min 50, Q₁ 70, median 75, Q₃ 85, max 95. Compare their performance and spread.
When comparing distributions, always state the median, quartiles and IQR. Comment on both centre and spread.
In a histogram, a class of width 5 has frequency 20. What is the frequency density?
💡 Frequency density = frequency ÷ class width = 20 ÷ 5 = 4.
For the data 1, 3, 5, 7, 9, 11, what is the median (Q₂)?
💡 The median is the average of the middle two values: (5 + 7) ÷ 2 = 6.
For the data 2, 4, 6, 8, 10, 12, what is the interquartile range?
💡 Q₁ = 4, Q₃ = 10. IQR = 10 − 4 = 6.
A cumulative frequency curve shows 200 students. The median is read at y = 100. True or false?
💡 The median is at the cumulative frequency of 200 ÷ 2 = 100.
Two box plots have the same median but different IQRs. What does this tell you?
💡 Median represents the centre. IQR represents the spread of the middle 50%.
Watch for these when working through the lesson.
These ideas fit closely with this lesson.