Box plots

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.