Box Plots

Overview

Welcome to your deep dive into Box Plots (Specification reference 5.3), a key data representation topic in OCR Level 2 Further Mathematics. Box plots, or box-and-whisker diagrams, are a powerful tool for visualising the spread and central tendency of a dataset. They provide a concise five-number summary that allows for rapid comparison between different groups. In your exam, you won't just be asked to draw them; you'll be expected to interpret them with a high degree of sophistication. This includes calculating and comparing medians and interquartile ranges (IQRs), and identifying skewness. Examiners use this topic to test your ability to move beyond simple calculations and make reasoned, contextual judgements about data. A typical question might present you with two box plots and ask you to 'Compare the distributions', a command word that requires a specific, two-part answer to secure all the marks. Mastering this topic provides a strong foundation for understanding statistical analysis and its connection to real-world data interpretation.

Key Concepts

Concept 1: The Five-Number Summary

A box plot is built from just five key values that summarise the entire dataset. To find them, your first step is always to arrange the data in ascending order. Failure to do so is a critical error from which you cannot recover in a question.

1. Minimum: The smallest value in the dataset.
2. Lower Quartile (Q1): The median of the lower half of the data. It marks the 25th percentile.
3. Median (Q2): The middle value of the entire dataset. It marks the 50th percentile.
4. Upper Quartile (Q3): The median of the upper half of the data. It marks the 75th percentile.
5. Maximum: The largest value in the dataset.

Example: Find the five-number summary for the data: 1, 8, 5, 3, 9, 8, 2, 10, 7

• Step 1: Order the data.
  1, 2, 3, 5, 7, 8, 8, 9, 10 (There are 9 values, so n=9)
• **Step 2: Find the Median (Q2).**The middle value is the (n+1)/2 term. (9+1)/2 = 5. The 5th value is 7. So, Median = 7.
• **Step 3: Find the Lower Quartile (Q1).**This is the median of the data to the left of the median: 1, 2, 3, 5. As there are an even number of values, we find the mean of the middle two: (2+3)/2 = 2.5. So, Q1 = 2.5.
• **Step 4: Find the Upper Quartile (Q3).**This is the median of the data to the right of the median: 8, 8, 9, 10. The mean of the middle two is (8+9)/2 = 8.5. So, Q3 = 8.5.
• Step 5: Identify Minimum and Maximum.
  Minimum = 1, Maximum = 10.

Concept 2: The Interquartile Range (IQR)

The IQR is a crucial measure of statistical spread or dispersion. It represents the range in which the middle 50% of the data lies. A smaller IQR indicates that the data points are clustered closely around the median, suggesting greater consistency. A larger IQR indicates the data is more spread out.

Formula: IQR = Upper Quartile (Q3) - Lower Quartile (Q1)

From our example above, the IQR would be 8.5 - 2.5 = 6.

In an exam, when comparing IQRs, you must use comparative language and link it to context. For instance, 'Group A has a smaller IQR than Group B, which means their performance was more consistent.' The word consistent is highly valued by examiners.

Concept 3: Identifying Skewness

Skewness describes the asymmetry of a distribution. A box plot gives clear visual clues about the skew.

• Symmetrical Distribution: The median is located exactly in the middle of the box (Q1 to Q3), and the whiskers are of roughly equal length. This indicates the data is evenly spread around the centre.
• Positive Skew (or Right-Skewed): The median is closer to the lower quartile (Q1). This often results in the right-hand side of the box and the right whisker being longer than the left. It suggests a larger number of lower values and a tail of higher values.
• Negative Skew (or Left-Skewed): The median is closer to the upper quartile (Q3). This often results in the left-hand side of the box and the left whisker being longer than the left. It suggests a larger number of higher values and a tail of lower values.

Examiners may ask you to 'Describe the skewness'. A good answer would be: 'The distribution is positively skewed, as the median is positioned closer to the lower quartile.'

Mathematical Relationships

• Median Position (from raw data): (n+1)/2 where n is the number of data points.
• Interquartile Range (IQR): Q3 - Q1 (Must memorise)
• Range: Maximum - Minimum (Must memorise)

Practical Applications

Box plots are used extensively in the real world to compare data sets quickly. For example:

• Business: A company might use box plots to compare the monthly sales figures of two different stores to see which is more successful and which is more consistent.
• Science: Biologists could compare the heights of plants grown with two different types of fertiliser.
• Finance: An analyst might compare the daily price fluctuations of two different stocks to assess volatility (consistency).

In all these cases, the goal is the same: use the median to compare the average and the IQR to compare the spread or consistency.