ST12: Histograms & Box Plots
Learn how to construct and interpret histograms with equal and unequal class widths, and box plots for comparing distributions in GCSE Statistics.
Histograms & Box Plots
Learn how to construct and interpret histograms with equal and unequal class widths, and box plots for comparing distributions in GCSE Statistics.
Key Fact: A histogram displays continuous data using bars with no gaps between them; the area of each bar (not the height) is proportional to the frequency.
Key Fact: Frequency density = frequency ÷ class width; the vertical axis of a histogram shows frequency density, not frequency.
Key Fact: When class widths are equal, the bar heights are proportional to the frequencies and frequency density is simply frequency ÷ class width.
Key Fact: When class widths are unequal, you must calculate frequency density for each bar; the height no longer equals the frequency.
Key Fact: The total area of all bars in a histogram equals the total frequency; you can find missing frequencies by setting up area equations.
Key Fact: A box plot (box-and-whisker diagram) shows five key values: minimum, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum.
Key Fact: The box in a box plot spans from Q1 to Q3, with a vertical line inside at the median; the whiskers extend to the minimum and maximum values.
Key Fact: Outliers on a box plot are values beyond 1.5 × IQR from the quartiles; they are plotted as individual points and the whisker stops at the last non-outlier value.
Key Fact: To compare distributions using box plots, comment on the median (central tendency), IQR (spread), and the position of the median within the box (skewness).
Key Fact: If the median is closer to Q1, the distribution is positively skewed; if closer to Q3, it is negatively skewed; if centred, it is approximately symmetric.
Key Fact: When drawing a histogram, always label the frequency density axis and mark the class boundaries on the horizontal axis.
Key Fact: To estimate the frequency of a range within a histogram, calculate the total area of the bars within that range.
📋 Key Vocabulary and Concepts
For Histograms & Box Plots, you must know:
- Histogram: A graphical display of continuous data using bars with no gaps, where the area of each bar is proportional to the frequency of its class interval.
- Frequency density: The height of a histogram bar, calculated as frequency ÷ class width; it allows comparison between classes of different widths.
- Box plot: A diagram showing the five-number summary of a dataset: minimum, lower quartile, median, upper quartile, and maximum, with a box from Q1 to Q3.
- Interquartile range (IQR): The difference between the upper quartile and lower quartile (Q3 – Q1), representing the spread of the middle 50% of the data.
- Outlier: A value that lies more than 1.5 × IQR below Q1 or above Q3; it is plotted as a separate point on a box plot.
- Whisker: The lines extending from the box to the minimum and maximum values (or to the last non-outlier value if outliers are present).
❓ Practice Questions
Q: A histogram has a class interval of width 5 with frequency 30. What is the frequency density?
Q: On a box plot, Q1 = 12, Q2 = 18, Q3 = 24. What is the IQR and what does the median's position tell you about skewness?
Q: A dataset has Q1 = 10, Q3 = 22. What are the outlier boundaries?
Q: In a histogram, the total area of bars is 200 and one bar has an area of 40. What is the frequency for that bar?
Q: Two box plots show the same median but different IQRs. What does this tell you?
✅ Answers
- Frequency density = 30 ÷ 5 = 6.
- IQR = 24 – 12 = 12. The median (18) is exactly midway between Q1 and Q3, so the distribution is approximately symmetric.
- IQR = 12. Lower boundary = 10 – 1.5 × 12 = –8. Upper boundary = 22 + 1.5 × 12 = 40. Values below –8 or above 40 are outliers.
- Since area is proportional to frequency, the frequency is 40 (assuming the scale factor is 1, i.e. area = frequency).
- The distributions have the same central value, but the one with the larger IQR has greater spread in the middle 50% of the data.
🎯 Exam Tips
- Always calculate frequency density before drawing a histogram — never use frequency directly on the y-axis unless all class widths are equal.
- When comparing box plots, write about median (location), IQR (spread), and overall range, and use comparative language such as 'higher' or 'more spread out'.
- On a box plot, the whisker does not extend to an outlier — it stops at the last data point within the 1.5 × IQR boundary.
- When estimating frequencies from a histogram, remember frequency = frequency density × class width (i.e. the area of the bar).
- If a histogram question gives you a missing frequency, set up an equation using total area = total frequency and solve.
📝 Exam Technique
GCSE Statistics Exam Tips — Histograms & Box Plots:
1. For Histograms & Box Plots questions, show every step of your working clearly — method marks count even if the final answer is wrong
2. Check your answer makes sense in context (estimation, units, reasonableness)
3. Use correct mathematical notation and state formulae before substituting values
4. If a Histograms & Box Plots question asks you to 'prove' or 'show', write a logical chain of reasoning with a conclusion line
5. For problem-solving, identify the topic first, then recall the relevant method
1. For Histograms & Box Plots questions, show every step of your working clearly — method marks count even if the final answer is wrong
2. Check your answer makes sense in context (estimation, units, reasonableness)
3. Use correct mathematical notation and state formulae before substituting values
4. If a Histograms & Box Plots question asks you to 'prove' or 'show', write a logical chain of reasoning with a conclusion line
5. For problem-solving, identify the topic first, then recall the relevant method
⚠️ Common Errors
✗ Using frequency on the y-axis of a histogram with unequal class widths ✓ The y-axis must show frequency density (= frequency ÷ class width); using frequency directly gives bars of incorrect height for unequal widths.
✗ Drawing the whisker all the way to an outlier on a box plot ✓ The whisker should end at the last data value within the 1.5 × IQR boundary; outliers are shown as separate points beyond the whisker.
✗ Saying a box plot with Q1=10, Q2=15, Q3=16 is negatively skewed because the right side of the box is shorter ✓ This is positively skewed — the median is closer to Q3, meaning the lower half of the middle 50% is more spread out, and the tail extends to the right.
✗ Calculating the area of a histogram bar as height × class width when the class boundaries are not stated correctly ✓ Use class width = upper boundary – lower boundary (e.g. for 10 ≤ x < 25, the width is 15). Always check boundary conventions carefully.
✍️ Model Answer
Full-Mark Response
The frequency table shows the times taken for 120 runners to complete a race. Draw a histogram. 0–10 min: frequency 15, 10–20 min: frequency 40, 20–40 min: frequency 45, 40–60 min: frequency 20
Calculate frequency density for each class: 0–10: width = 10, FD = 15 ÷ 10 = 1.5 10–20: width = 10, FD = 40 ÷ 10 = 4.0 20–40: width = 20, FD = 45 ÷ 20 = 2.25 40–60: width = 20, FD = 20 ÷ 20 = 1.0 Draw the histogram with class boundaries on the x-axis (0, 10, 20, 40, 60) and frequency density on the y-axis. Each bar has no gap and its height equals the frequency density. Check: total area = (1.5 × 10) + (4.0 × 10) + (2.25 × 20) + (1.0 × 20) = 15 + 40 + 45 + 20 = 120
📊 AO Deep Dive
Assessment Objective Analysis
AO1 (Knowledge & Understanding): Demonstrate knowledge and understanding of histograms & box plots, including data collection, presentation and calculation techniques relevant to Edexcel 1ST0 & AQA 8382.
AO2 (Application): Apply knowledge and understanding of histograms & box plots to interpret data, reason statistically and draw conclusions in context.
AO3 (Evaluation): Evaluate statistical methods and conclusions, assessing appropriateness, reliability, validity and bias through the statistical enquiry cycle.
📝 Exam Questions by Topic
- AQA Past Papers - Topic Questions -- AQA
- Edexcel Past Papers - Topic Questions -- Edexcel
- CCEA Past Papers - Topic Questions -- CCEA
🎬 Video Resources
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