Box plots

Topic summary

Boxplots are graphical representations of a dataset’s distribution. They display key statistics, including the minimum, lower quartile (\(Q_1\)), median, upper quartile (\(Q_3\)), and maximum, along with any outliers. Boxplots are useful for visualising and comparing distributions.

1. Drawing a Boxplot:

To construct a boxplot, follow these steps:

1. Organise the data: Arrange the data in ascending order and calculate the five key values:
   • Minimum: The smallest value (excluding outliers).
   • \(Q_1\): The lower quartile (25th percentile).
   • Median: The middle value (50th percentile).
   • \(Q_3\): The upper quartile (75th percentile).
   • Maximum: The largest value (excluding outliers).
2. Identify outliers: Calculate the interquartile range (IQR): \[ \text{IQR} = Q_3 – Q_1 \] Use the formulas below to find thresholds for outliers:
   • Lower threshold: \(Q_1 – 1.5 \times \text{IQR}\).
   • Upper threshold: \(Q_3 + 1.5 \times \text{IQR}\).

   Any data points outside these thresholds are considered outliers.

3. Plot the key values: Draw a number line, marking the minimum, \(Q_1\), median, \(Q_3\), and maximum. Connect \(Q_1\), median, and \(Q_3\) with a box, and draw whiskers from the box to the minimum and maximum (excluding outliers).
4. Plot outliers: Represent outliers as individual points beyond the whiskers.

2. Interpreting a Boxplot:

Boxplots summarise a dataset’s key characteristics:

• The box represents the middle 50% of the data, bounded by \(Q_1\) and \(Q_3\).
• The median line within the box indicates the centre of the distribution.
• The whiskers extend to the smallest and largest non-outlier values, showing the range.
• Outliers, plotted as individual points, highlight unusual data values.

Key features to consider when interpreting a boxplot include:

• The spread of the data (indicated by the length of the box and whiskers).
• The presence and location of outliers.
• Skewness: If the median is closer to \(Q_1\) or \(Q_3\), the data may be skewed.

3. Comparing Boxplots:

When comparing multiple boxplots, look for differences in:

• Medians: Indicate differences in central tendency.
• Spread: Compare the lengths of boxes and whiskers to identify variations in variability.
• Outliers: Examine the number and positions of outliers for each dataset.
• Skewness: Look for asymmetry in the box and whiskers to assess skewness in the data.

4. Example:

Consider the following dataset: \(2, 4, 5, 7, 9, 12, 14, 18, 22\).

• Step 1: Arrange the data in ascending order (already arranged).
• Step 2: Find the key values:
  • Minimum: \(2\)
  • \(Q_1: 5\)
  • Median: \(9\)
  • \(Q_3: 14\)
  • Maximum: \(22\)
• Step 3: Calculate the IQR: \(Q_3 – Q_1 = 14 – 5 = 9\).
• Step 4: Find thresholds for outliers:
  • Lower threshold: \(5 – 1.5 \times 9 = -8.5\).
  • Upper threshold: \(14 + 1.5 \times 9 = 27.5\).
• Step 5: Identify outliers: There are no outliers, as all data points lie between \(-8.5\) and \(27.5\).

Plot the key values and connect them with a box and whiskers.

5. Summary:

• Boxplots visualise the distribution of a dataset, highlighting key statistics and outliers.
• Use the IQR method to identify outliers and ensure they are represented on the plot.
• Compare boxplots to analyse differences in central tendency, spread, and outliers between datasets.