D3a — AQA GCSE Statistics
Test yourself on D3a with AQA GCSE practice questions.
Start free
7 days Premium · Then free forever · No card, no charge
Your focus
- Calculate different measures of spread:
D3a exam tips
Quick Revision Summary (Key Takeaway)
D3a in AQA GCSE Statistics covers the construction, interpretation and comparison of box plots (box-and-whisker diagrams) to summarise and analyse distributions. You must be able to find the five-figure summary (minimum, lower quartile, median, upper quartile, maximum), draw accurate box plots, and use them to compare data sets in terms of median, interquartile range, range and skewness.
Topic Overview
D3a is part of the AQA GCSE Statistics specification focusing on box plots (box-and-whisker diagrams). This topic teaches you how to summarise a data set using five key values: the minimum, lower quartile, median, upper quartile and maximum. You will learn to construct box plots accurately and interpret them to compare distributions.
Box plots are essential for comparing two or more data sets visually. They show both a measure of average (median) and measures of spread (interquartile range and range), making them powerful tools for analysing real-world data such as test scores, waiting times or product lifetimes. This topic also links to cumulative frequency graphs and quartiles, which are often examined together.
Key Concepts
- →The five-figure summary consists of: minimum value, lower quartile (LQ), median, upper quartile (UQ) and maximum value.
- →The interquartile range (IQR) = UQ - LQ. It measures the spread of the middle 50% of the data and is not affected by extreme values (outliers).
- →To find quartiles, first order the data. For an odd number of values, exclude the median when splitting into halves. For an even number, split into two equal halves.
- →A box plot is drawn on a scale with a box from LQ to UQ, a line inside the box at the median, and whiskers from the box to the minimum and maximum.
- →When comparing box plots, comment on the median (average) and the interquartile range or range (spread/consistency), always in the context of the data.
Examiner Tips
- 💡Always show your method for finding quartiles, especially when the data set is large. This can earn method marks even if you make an arithmetic error.
- 💡When drawing a box plot, use a ruler and ensure the scale is clearly labelled. The box plot should be drawn accurately to scale.
- 💡In comparison questions, use the phrase 'on average' when referring to the median and 'more consistent' or 'more spread out' when referring to the IQR or range. Always relate your answer back to the context.
Common Mistakes
- Students often think the lower quartile is the minimum and the upper quartile is the maximum. Correction: The quartiles divide the data into four equal parts; the minimum and maximum are the extremes, not quartiles.
- Students sometimes forget to order the data before finding quartiles, leading to incorrect values. Correction: Always sort the data in ascending order first.
- When comparing box plots, students may only compare medians and ignore spread. Correction: A full comparison must mention both a measure of average (median) and a measure of spread (IQR or range).
Revision Plan
- 1Day 1: Revise how to find the median, lower quartile and upper quartile from a list of data. Practice with both odd and even numbers of values.
- 2Day 2: Learn the five-figure summary and practice finding it from raw data and from cumulative frequency graphs.
- 3Day 3: Practice drawing box plots accurately on graph paper, ensuring correct scale and labelling.
- 4Day 4: Work through comparison questions. Focus on writing full sentences that compare median and IQR/range in context.
- 5Day 5: Complete past paper questions on box plots. Mark your answers using the mark scheme and note where you lost marks.
Exam Question Types
- 📋Finding the five-figure summary from a list of data or a cumulative frequency graph. Advice: Order the data, find the median, then find the quartiles by splitting the data into halves.
- 📋Drawing a box plot from given summary statistics or raw data. Advice: Use a ruler, label the scale clearly, and ensure the box and whiskers are drawn to scale.
- 📋Comparing two box plots in context. Advice: Compare medians (average) and interquartile ranges (spread/consistency), and always refer back to the context.
- 📋Interpreting box plots to identify outliers or skewness. Advice: Remember that a longer whisker indicates skewness; outliers may be shown as separate points.
Command Word Expectations (AQA)
Calculate
You must work out a numerical answer, showing clear working. For example, 'Calculate the interquartile range' requires you to subtract LQ from UQ and state the result.
Compare
You must describe similarities and differences between two sets of data. For box plots, compare medians and interquartile ranges (or ranges) in context, using comparative language such as 'higher', 'lower', 'more consistent'.
Draw
You must construct an accurate diagram using a ruler and a suitable scale. For a box plot, ensure the box, median line and whiskers are correctly positioned and the scale is labelled.
How Students Lose Marks (Examiner Pitfalls)
Pitfall: Students often confuse the lower quartile (LQ) with the minimum value, or the upper quartile (UQ) with the maximum, especially when the data set is small. They also forget to order the data before finding quartiles.
❌ Weak Answer (Loses Marks):The lower quartile is the smallest number in the list, and the upper quartile is the largest number.
Example improved answer:The lower quartile is the median of the lower half of the ordered data, and the upper quartile is the median of the upper half. For example, for the ordered data 2, 4, 5, 7, 8, 9, 11, the median is 7, the lower quartile is 4, and the upper quartile is 9. The minimum is 2 and the maximum is 11.
Examiner Tip: Always order the data first. For an odd number of values, exclude the median when finding the quartiles. For an even number, split the data into two equal halves and find the median of each half. Remember: minimum and maximum are not quartiles.
Pitfall: When comparing two box plots, students often only compare medians and ignore the spread (interquartile range or range). They also fail to relate the comparison back to the context of the question.
❌ Weak Answer (Loses Marks):Group A has a higher median than Group B, so Group A is better.
Example improved answer:Group A has a higher median (e.g. 15 compared to 10), indicating that on average Group A scored higher. However, Group A also has a larger interquartile range (e.g. 8 compared to 4), meaning the middle 50% of Group A's scores are more spread out, so Group A is less consistent than Group B. Therefore, Group A tends to score higher but is more variable.
Examiner Tip: Always compare both a measure of average (median) and a measure of spread (IQR or range). Use the phrase 'on average' for median and 'more consistent' or 'more spread out' for IQR. Always refer back to the context (e.g. 'scores', 'waiting times').
Step-by-Step Worked Solutions
Question: The number of goals scored by a football team in 11 matches is: 3, 1, 2, 0, 4, 2, 1, 3, 5, 2, 1. (a) Find the five-figure summary. (b) Draw a box plot to represent the data.
- 1.Step 1: Order the data: 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5.
- 2.Step 2: Identify the minimum (0) and maximum (5).
- 3.Step 3: Find the median (middle value). There are 11 values, so the median is the 6th value: 2.
- 4.Step 4: Find the lower quartile. The lower half (excluding the median) is 0, 1, 1, 1, 2. The median of these 5 values is the 3rd value: 1.
- 5.Step 5: Find the upper quartile. The upper half is 2, 3, 3, 4, 5. The median is the 3rd value: 3.
- 6.Step 6: Five-figure summary: Minimum = 0, LQ = 1, Median = 2, UQ = 3, Maximum = 5.
- 7.Step 7: Draw a horizontal scale covering 0 to 5. Draw a box from LQ (1) to UQ (3) with a vertical line at the median (2). Draw whiskers from the box to the minimum (0) and maximum (5).
Final Answer: Five-figure summary: 0, 1, 2, 3, 5. Box plot drawn with box from 1 to 3, median line at 2, whiskers to 0 and 5.
Question: Two box plots show the waiting times (in minutes) at two different GP surgeries. Surgery A: min=5, LQ=10, median=15, UQ=20, max=30. Surgery B: min=8, LQ=12, median=18, UQ=22, max=25. Compare the waiting times at the two surgeries.
- 1.Step 1: Compare medians. Surgery A median = 15, Surgery B median = 18. Surgery B has a higher median waiting time, so on average patients wait longer at Surgery B.
- 2.Step 2: Compare interquartile ranges. Surgery A IQR = 20 - 10 = 10. Surgery B IQR = 22 - 12 = 10. Both have the same IQR, so the middle 50% of waiting times are equally spread.
- 3.Step 3: Compare ranges. Surgery A range = 30 - 5 = 25. Surgery B range = 25 - 8 = 17. Surgery A has a larger range, indicating more variability overall, with some very long waits.
- 4.Step 4: Write a conclusion in context. Surgery B has a higher typical waiting time but is more consistent overall (smaller range). Surgery A has a lower typical wait but occasionally has very long waits.
Final Answer: Surgery B has a higher median waiting time (18 vs 15) but a smaller range (17 vs 25), so waits are generally longer but more consistent. Surgery A has a lower median but greater overall variability.
Active Recall Memory Test
What are the five values needed to draw a box plot?
Key Fact: Minimum, lower quartile (LQ), median, upper quartile (UQ), maximum.
How do you calculate the interquartile range?
Key Fact: IQR = Upper Quartile - Lower Quartile.
What does a larger interquartile range indicate about a data set?
Key Fact: A larger IQR indicates that the middle 50% of the data is more spread out, so the data is more variable or less consistent.
When comparing two box plots, what two features should you always compare?
Key Fact: The median (a measure of average) and the interquartile range or range (a measure of spread).
Frequently Asked Questions
How do I find the lower quartile and upper quartile?
First, order the data from smallest to largest. Find the median. For an odd number of values, exclude the median and split the remaining data into two halves. The lower quartile is the median of the lower half, and the upper quartile is the median of the upper half. For an even number of values, split the data into two equal halves and find the median of each half.
What is the difference between range and interquartile range?
The range is the difference between the maximum and minimum values, so it includes all data and can be affected by outliers. The interquartile range (IQR) is the difference between the upper quartile and lower quartile, so it only considers the middle 50% of the data and is not affected by extreme values. The IQR is often a better measure of spread when there are outliers.
How do I compare two box plots in an exam?
Start by comparing the medians: state which is higher or lower and what that means in context (e.g. 'on average, Group A scored higher'). Then compare the interquartile ranges or ranges: state which is larger and what that means about consistency or spread. Always use comparative language and refer back to the context of the question.
Can a box plot show outliers?
Yes, box plots can show outliers as individual points beyond the whiskers. In AQA GCSE Statistics, you may be asked to identify outliers using the rule that any value more than 1.5 times the IQR below the lower quartile or above the upper quartile is an outlier. However, the standard box plot in D3a typically uses the minimum and maximum as whiskers unless outliers are specified.
What does it mean if the median is closer to the lower quartile than the upper quartile?
If the median is closer to the lower quartile, the distribution is positively skewed (skewed to the right). This means there is a longer tail on the right side of the distribution, indicating a few higher values. Conversely, if the median is closer to the upper quartile, the distribution is negatively skewed (skewed to the left).
How do I draw a box plot accurately?
Use a ruler and a suitable scale that covers the full range of the data. Draw a horizontal line and mark the scale clearly. Draw a box from the lower quartile to the upper quartile, with a vertical line inside the box at the median. Then draw whiskers from the ends of the box to the minimum and maximum values. Ensure all lines are straight and the plot is neat.