Reading a box plot
A box plot shows: minimum = 20, Q1 = 30, median = 40, Q3 = 55, maximum = 80. Describe the distribution.
The box contains the middle 50% of data. The whiskers show the range.
Year 11
Formal data displays and comparison
Read box plots, cumulative graphs and histograms fluently, then justify comparisons with precise language.
Lesson overview
Statistics and data — displays and interpretation
Box plots summarise data using quartiles: lower quartile (Q1), median (Q2), upper quartile (Q3), and range. Cumulative frequency curves show cumulative totals and allow reading of quartiles and percentiles. Histograms use frequency density (frequency ÷ class width) on the y-axis to correctly represent grouped data with unequal class widths. Comparing two distributions requires careful language: reference median (central tendency), interquartile range (spread), and outliers. Always connect statistical language to the context of the data.
Reading from a cumulative frequency curve
150 items are grouped as: 0-10 (20), 10-20 (30), 20-30 (40), 30-40 (35), 40-50 (25). Find the median and Q1.
Cumulative frequency is the running total. Find the position, then read horizontally to the curve, then down to the axis.
Understanding histograms with frequency density
A histogram shows time (minutes) grouped as: 0-10 (15), 10-20 (30), 20-40 (40), 40-60 (10). Class 10-20 has frequency 30 and width 10.
In a histogram with unequal class widths, height = frequency density = frequency ÷ width.
Comparing two distributions
Two classes' test scores: Class A has median 65 and IQR 15. Class B has median 60 and IQR 10. Compare them.
Use 'median', 'IQR', 'range' and 'outliers' to justify comparisons. Connect to the context.
Worked example — comparing two data sets
Two shops' daily sales (in £100s): Shop A has min=20, Q1=50, median=65, Q3=80, max=95. Shop B has min=25, Q1=55, median=62, Q3=72, max=90. Compare the shops.
- Compare medians: Shop A (65) has higher median than Shop B (62). Shop A's typical daily sales are higher.
- Compare IQRs: Shop A (30) has larger IQR than Shop B (17). Shop A's sales vary more widely; Shop B is more consistent.
- Conclusion: Shop A generally makes higher sales but with more variability. Shop B has more predictable sales.
- In context: Shop A is more successful but riskier. Shop B is more reliable but lower-earning.
Try it
Use the examples carefully, then choose the answer.
Question 1
From a box plot, the median is 50 and the upper quartile is 75. What percentage of data lies between 50 and 75?
💡 The median divides at 50%. The upper quartile is at 75%, so between them is 25%.
Question 2
In a histogram, a class with width 5 has 20 items. What is the frequency density?
💡 Frequency density = frequency ÷ width = 20 ÷ 5 = 4.
Question 3
From a cumulative frequency graph, there are 120 total items. At what cumulative frequency do you find Q3?
💡 Q3 = of total = × 120 = 90.
Question 4
Two data sets: Set A has range 50, Set B has range 30. What can you conclude?
💡 Range measures spread. Larger range = more spread.
Question 5
A histogram bar covers 10–15 kg with height 0.4. How many items are in this class?
💡 Frequency = frequency density × width = 0.4 × 5 = 2.
Common mistakes
Watch for these when working through the lesson.
- Confusing frequency with frequency density in histograms; always use frequency density = frequency ÷ width.
- Reading the wrong position on a cumulative frequency curve; the curve shows cumulative total, not individual class frequency.
- Forgetting that Q1 = 25th percentile, median = 50th, Q3 = 75th, not arbitrary positions.
- Making comparisons without using precise statistical language ('median', 'IQR', 'range') or connecting to context.
Related topics
These ideas fit closely with this lesson.
- Mean, median, mode and measures of spread
- Frequency tables and grouped data
- Probability and inference
- Sampling and data collection