Quartiles, Box Plots and Comparing Populations

Worked Example: Graph Estimation

A cumulative frequency graph shows the masses, in grams, of 120 apples. The total frequency ($n$) is 120. Calculate an estimate for the interquartile range.

Step 1: Find the Lower Quartile ($Q_1$) position and value.

• Position: $\frac{120}{4} = 30$
• Read across from 30 on the cumulative frequency (y) axis to the curve, and down to the mass (x) axis.
• Estimate from graph: Lower Quartile = $82$ g

Step 2: Find the Upper Quartile ($Q_3$) position and value.

• Position: $\frac{3 \times 120}{4} = 90$
• Read across from 90 on the y-axis to the curve, and down to the x-axis.
• Estimate from graph: Upper Quartile = $106$ g

Step 3: Calculate the Interquartile Range.

• $\text{IQR} = 106 - 82 = 24$ g

Constructing Box Plots

A box plot (or box-and-whisker diagram) is a visual graph drawn on a linear scale that represents the five-number summary of a dataset.

Step-by-Step: Drawing a Box Plot from Raw Data To draw a box plot for the data set: $3, 7, 8, 12, 14, 15, 19$

1. Identify the extremes: Minimum = $3$, Maximum = $19$.
2. Find the Median ($Q_2$): The middle value is $12$.
3. Find the Quartiles: The median of the lower half $\{3, 7, 8\}$ is $Q_1 = 7$. The median of the upper half $\{14, 15, 19\}$ is $Q_3 = 15$.
4. Draw the plot:
   • Mark the five values ($3, 7, 12, 15, 19$) with vertical lines above a numbered scale.
   • Draw a rectangular box from $Q_1$ to $Q_3$.

An outlier is an extreme value that does not fit the overall pattern. On Higher Tier papers, an outlier is any value $> Q_3 + (1.5 \times \text{IQR})$ or $< Q_1 - (1.5 \times \text{IQR})$. These are marked with a cross ($\times$) and the whisker stops at the next nearest "normal" value.

Comparing Distributions

When asked to compare two populations using box plots, you must provide a paired comparison of location (average) and dispersion (spread).

Example: Side-by-Side Comparison of Test Scores Imagine two box plots, Class A and Class B, are drawn on the same scale.

Standard Comparison Structure:

1. Compare the Medians (Location): "The median for Class B ($55$) is higher than Class A ($48$), meaning Class B performed better on average."
2. Compare the IQRs (Dispersion): "The IQR for Class A ($25$) is smaller than Class B ($26$), meaning Class A's scores are slightly more consistent."

Note: Always use the context of the question (e.g., 'test scores', 'running times') to gain full marks.