Box Plots Quizzes

Box Plots Quiz 1

Difficulty: Foundation

Curriculum: GCSE

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Box Plots Quiz 1

Difficulty: Higher

Curriculum: GCSE

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Introduction

Box plots, also called box-and-whisker diagrams, are a graphical method for summarising data distributions. They show the median, quartiles, and extremes of a dataset, making it easier to understand the spread, centre, and skewness of data. Box plots are essential for GCSE Maths statistics and help students visualise distributions quickly.

Core Concepts

What is a Box Plot?

A box plot represents a dataset using five key statistics:

  • Minimum: The smallest data point (excluding outliers).
  • Q1 (Lower Quartile): The 25th percentile, dividing the lowest 25% of data.
  • Median: The middle value (50th percentile).
  • Q3 (Upper Quartile): The 75th percentile, dividing the highest 25% of data.
  • Maximum: The largest data point (excluding outliers).

Sometimes, box plots also show outliers, which are unusually high or low values compared to the rest of the dataset.

Key Features of a Box Plot

  • Box: Extends from Q1 to Q3.
  • Median line: Drawn inside the box.
  • Whiskers: Lines extending from the box to the minimum and maximum values.
  • Outliers: Plotted individually beyond the whiskers.

Why Use Box Plots?

  • Visualise the spread and symmetry of data.
  • Compare distributions between datasets.
  • Identify skewness and potential outliers.

Rules & Steps for Drawing a Box Plot

  1. Order your dataset from smallest to largest.
  2. Find the minimum and maximum values.
  3. Calculate Q1, median, and Q3:
    • Median divides the dataset into two halves.
    • Q1 = median of lower half.
    • Q3 = median of upper half.
  4. Draw a number line covering the range of your data.
  5. Draw a box from Q1 to Q3 and a line at the median.
  6. Draw whiskers from the box to the minimum and maximum values.
  7. Plot any outliers individually beyond the whiskers.

Worked Examples

Example 1: Basic Box Plot

Dataset: 4, 7, 8, 9, 10, 12, 15

  • Step 1: Order the data (already ordered).
  • Step 2: Minimum = 4, Maximum = 15
  • Step 3: Median = 9
  • Step 4: Lower half: 4, 7, 8 → Q1 = 7
  • Step 5: Upper half: 10, 12, 15 → Q3 = 12

Construct the box plot:

  • Box from Q1 = 7 to Q3 = 12
  • Median line at 9
  • Whiskers to minimum = 4 and maximum = 15

Example 2: Identifying Outliers

Dataset: 3, 4, 5, 6, 7, 8, 9, 15

Step 1: Calculate quartiles:

  • Median = (6 + 7)/2 = 6.5
  • Lower half: 3, 4, 5, 6 → Q1 = (4+5)/2 = 4.5
  • Upper half: 7, 8, 9, 15 → Q3 = (8+9)/2 = 8.5

Step 2: Calculate Interquartile Range (IQR):

$$ \text{IQR} = Q3 - Q1 = 8.5 - 4.5 = 4 $$

Step 3: Determine outlier boundaries:

  • Lower bound = Q1 - 1.5 × IQR = 4.5 - 6 = -1.5 → no lower outliers
  • Upper bound = Q3 + 1.5 × IQR = 8.5 + 6 = 14.5 → 15 is an outlier

Construct the box plot with whiskers ending at 9 (maximum non-outlier) and plot 15 individually as an outlier.

Example 3: Comparing Two Box Plots

Dataset A: 5, 6, 7, 8, 9, 10

Dataset B: 2, 4, 5, 6, 7, 12

  • Calculate medians, Q1, Q3, minimum, maximum.
  • Draw box plots for both datasets on the same scale.
  • Compare spread: Dataset B has a larger range and possible outlier (12).
  • Interpret skewness and variability visually.

Common Mistakes

  • Incorrectly calculating Q1, median, or Q3, especially with odd or even number of data points.
  • Forgetting to check for outliers using 1.5 × IQR rule.
  • Misplacing whiskers – whiskers should extend to min/max excluding outliers.
  • Using box plots for categorical data instead of continuous data.

Applications

Box plots are widely used in exams and real-life scenarios:

  • Comparing test scores between two classes or schools.
  • Showing variability in temperature or rainfall.
  • Detecting unusual values in sales or scientific data.

Strategies & Tips

  • Always order the dataset before calculating quartiles.
  • Double-check medians for both halves when calculating Q1 and Q3.
  • Use the IQR rule to identify outliers and plot them separately.
  • Label all axes clearly and use a consistent scale.
  • Practice interpreting box plots to describe centre, spread, and skewness in exam questions.

Summary & Encouragement

Box plots summarise data efficiently, showing centre, spread, and outliers at a glance. Key points to remember:

  • Identify minimum, Q1, median, Q3, and maximum.
  • Use IQR to detect outliers.
  • Whiskers extend to minimum/maximum excluding outliers.
  • Box plots allow easy comparison between datasets.

Practice constructing box plots from various datasets and interpreting them to improve your statistical reasoning. Complete the quizzes to strengthen your understanding of box plots and their applications in GCSE Maths!