GCSE Maths Statistics and Probability - How to Solve: Box Plots, Cumulative Frequency, and Histograms

How to Solve: Box Plots, Cumulative Frequency, and Histograms

GCSE / A-Level (Physics, Chemistry, Biology) – Complete Guide

Introduction

"Mastering box plots, cumulative frequency, and histograms lets you analyse real-world data—like drug trial results, reaction times, or species distributions—and earn up to 12 marks in your GCSE/A-Level exams. One question on these topics can be the difference between a 6 and a 7, or a B and an A!"

WHAT YOU NEED TO KNOW FIRST

Before diving in, make sure you understand: 1. Basic statistics – Mean, median, mode, range, and quartiles. 2. Frequency tables – How to read and construct them. 3. Grouped data – Class intervals and how to find midpoints.

If you’re shaky on any of these, pause and review them first.

KEY TERMS & FORMULAS

Key Terms

Formulas

1. Median (Q2) Position
2. Formula: (n + 1) / 2 (where n = number of data points)

3. MEMORISE THIS – Used to find the median in a list of numbers.

4. Quartile Positions

5. Q1 Position: (n + 1) / 4
6. Q3 Position: 3(n + 1) / 4

7. MEMORISE THIS – Used to find quartiles in ordered data.

8. Frequency Density (for Histograms)

9. Formula: Frequency / Class Width

10. MEMORISE THIS – Used to calculate bar height in histograms.

11. Outlier Boundaries

12. Lower Bound: Q1 – 1.5 × IQR
13. Upper Bound: Q3 + 1.5 × IQR
14. MEMORISE THIS – Used to identify outliers in box plots.

STEP-BY-STEP METHOD

How to Draw a Box Plot

Step 1: Order the data from smallest to largest. Step 2: Find the minimum, Q1, median (Q2), Q3, and maximum. Step 3: Draw a number line covering the full range of data. Step 4: Plot the five key points (min, Q1, Q2, Q3, max) above the line. Step 5: Draw a box from Q1 to Q3. Step 6: Draw a vertical line inside the box at the median (Q2). Step 7: Draw "whiskers" from Q1 to the minimum and from Q3 to the maximum. Step 8: Check for outliers (points outside 1.5 × IQR). If present, mark them with a cross and adjust whiskers to the next closest value.

How to Draw a Cumulative Frequency Graph

Step 1: Make a cumulative frequency table (add each frequency to the previous total). Step 2: Plot the upper boundary of each class against its cumulative frequency. Step 3: Join points with a smooth curve or straight lines. Step 4: Use the graph to estimate:
- Median (at 50% of total frequency)
- Quartiles (at 25% and 75% of total frequency)
- Interquartile Range (IQR) = Q3 – Q1

How to Draw a Histogram

Step 1: Calculate class width (upper bound – lower bound). Step 2: Calculate frequency density = Frequency / Class Width. Step 3: Draw a number line covering all class intervals. Step 4: For each class, draw a bar with:
- Width = class width
- Height = frequency density Step 5: Label axes:
- x-axis = class intervals
- y-axis = frequency density

WORKED EXAMPLES

Example 1 – Basic Box Plot

Data: 3, 5, 7, 8, 9, 10, 12, 15 Step 1: Ordered data is already given. Step 2:
- Minimum = 3
- Q1 = 6 (average of 5 and 7)
- Median (Q2) = 8.5 (average of 8 and 9)
- Q3 = 11 (average of 10 and 12)
- Maximum = 15 Step 3-8: Draw the box plot (see diagram). What we did and why: We found the five key points and plotted them to show the spread of data.

Example 2 – Medium Cumulative Frequency Graph

Data Table: | Time (s) | Frequency | Cumulative Frequency | |----------|-----------|----------------------| | 0-10 | 5 | 5 | | 10-20 | 8 | 13 | | 20-30 | 12 | 25 | | 30-40 | 6 | 31 |

Step 1: Cumulative frequencies are already calculated. Step 2: Plot points at (10,5), (20,13), (30,25), (40,31). Step 3: Join with straight lines. Step 4: Estimate median (at 15.5th value) ≈ 22s. What we did and why: We used cumulative frequency to find the median without listing all data.

Example 3 – Exam-Style Histogram

Question: A biologist records plant heights (cm). Draw a histogram. | Height (cm) | Frequency | |-------------|-----------| | 0-10 | 4 | | 10-20 | 12 | | 20-30 | 18 | | 30-50 | 16 |

Step 1: Class widths = 10, 10, 10, 20. Step 2: Frequency densities = 0.4, 1.2, 1.8, 0.8. Step 3-5: Draw bars with correct heights and widths. What we did and why: We adjusted for unequal class widths by using frequency density.

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP

"Right, listen up—this is your last-minute checklist. For box plots, remember the five key points: min, Q1, median, Q3, max. For cumulative frequency, plot the upper bound and read off the median at 50%. For histograms, frequency density = frequency ÷ class width—never just plot frequency! Outliers? 1.5 × IQR above Q3 or below Q1. Double-check your axes, label everything, and you’ll smash those 12 marks. Now go ace that exam!"