Quartiles, Box Plots and Comparing Distributions

The difference between these two values is called the Inter-quartile Range (IQR). Because the IQR only measures the middle 50% of the data, it is a highly reliable measure of spread that is not affected by outliers (extreme values).

$$IQR = Q_3 - Q_1$$

To find the quartiles from a discrete list, you first order the data and use the formula $\frac{n+1}{4}$ to find the position, where $n$ is the total number of values. If the position is a decimal (e.g., the $2.5^{\text{th}}$ position), you take the midpoint of the two surrounding values.

Worked Example: Calculate the IQR for the following data set: $4, 7, 9, 13, 15, 18, 19, 20, 23$.

Step 1: Count the values to find $n$.

• There are 9 values, so $n = 9$.

Step 2: Find the Lower Quartile ($Q_1$).

• Position of $Q_1$ = $\frac{9+1}{4} = 2.5^{\text{th}}$ value.
• The $2.5^{\text{th}}$ value is halfway between the $2^{\text{nd}}$ value ($7$) and $3^{\text{rd}}$ value ($9$).
• $Q_1 = \frac{7+9}{2} = 8$

Step 3: Find the Upper Quartile ($Q_3$).

• Position of $Q_3$ = $\frac{3(9+1)}{4} = 7.5^{\text{th}}$ value.
• The $7.5^{\text{th}}$ value is halfway between the $7^{\text{th}}$ value ($19$) and $8^{\text{th}}$ value ($20$).
• $Q_3 = \frac{19+20}{2} = 19.5$

Step 4: Calculate the IQR.

• $IQR = Q_3 - Q_1$
• $IQR = 19.5 - 8 = 11.5$

Identifying Outliers (Higher Tier)

An outlier is a data point that is significantly different from the rest of the dataset. For AQA Higher Tier, you must calculate exactly where the boundary for an outlier begins using the $1.5 \times IQR$ rule.

• Lower Bound: $Q_1 - (1.5 \times IQR)$
• Upper Bound: $Q_3 + (1.5 \times IQR)$

Any value in the dataset that falls below the lower bound or above the upper bound is officially classified as an outlier. On a graph, outliers are plotted with a cross (X) or a plus (+).

Constructing and Interpreting Box Plots

A Box Plot (Box-and-Whisker Diagram) is a graphical representation used to show the spread and central tendency of a dataset. To draw one, you need a Five-Number Summary: the Minimum value, $Q_1$, Median ($Q_2$), $Q_3$, and Maximum value.

To construct a box plot, follow these steps strictly:

1. First, order your data and calculate the five-number summary.
2. Next, draw a horizontal axis with a consistent, linear scale that covers your minimum and maximum values.
3. Then, plot five short vertical lines above the scale at the exact positions of your Minimum, $Q_1$, Median, $Q_3$, and Maximum.
4. After that, connect the tops and bottoms of the $Q_1$ and $Q_3$ lines with horizontal lines to form the central box. This box represents the IQR.
5. Finally, draw horizontal lines called whiskers extending from the middle of the box edges out to the Minimum and Maximum lines.

If your dataset contains an outlier, do NOT extend the whisker to the outlier. Instead, mark the outlier with a cross, and end the whisker at the next highest (or lowest) value that is not an outlier.

Reading and Interpreting a Box Plot: To extract the five-number summary and other specific values from an existing box plot scale, follow these steps:

1. Check the scale: Determine exactly what one small square on the horizontal axis represents (e.g., $1$ unit, $2$ units, or $5$ units).
2. Minimum: Trace a line straight down from the end of the left whisker to the axis and read the value.
3. Lower Quartile ($Q_1$): Trace a line down from the left edge of the central box.
4. Median: Trace a line down from the vertical line inside the box.
5. Upper Quartile ($Q_3$): Trace a line down from the right edge of the central box.
6. Maximum: Trace a line down from the end of the right whisker.

If there are crosses (X) or pluses (+) plotted beyond the whiskers, track them down to the axis to read the values of the outliers. You can then calculate the IQR of the distribution by subtracting the $Q_1$ value from the $Q_3$ value.

Analysing and Comparing Distributions

When asked to compare two data distributions, you must always provide two distinct comparisons: one for central tendency (usually the median) and one for spread (the IQR).

Each comparison requires two sentences. The first sentence must state the numerical difference, and the second must explain what this means in context. A smaller IQR means the data is more stable, which is described as having higher consistency.

Statistical Measure	Numerical Comparison (Sentence 1)	Contextual Meaning (Sentence 2)
Central Tendency (Median)	The median time for Group A ($45$ seconds) is lower than Group B ($55$ seconds).	Group A was faster on average.
Spread (IQR)	Group A has a smaller IQR ($10$ seconds) than Group B ($18$ seconds).	Group A's times are more consistent (less varied).

You can also comment on the shape of the box plot. If the median line is closer to $Q_1$, the data is positively skewed. If the median line is closer to $Q_3$, the data is negatively skewed.