Representing and Comparing Data Distributions

Constructing Polygons and Cumulative Frequency Curves

To draw a frequency polygon, points are plotted at the midpoint of the class interval. The midpoint is found by adding the lower and upper bounds and dividing by two: $\frac{\text{Lower Bound} + \text{Upper Bound}}{2}$.

These points are joined with straight lines to create an "open" polygon. You should not join the first point to the last point, nor to the x-axis, unless the frequency is exactly 0 at that boundary.

In contrast, cumulative frequency graphs are plotted at the upper boundary of the class interval. The points are joined with a smooth, S-shaped curve (ogive) rather than straight lines.

Histograms (Higher Tier)

Why do some bar-style graphs have unequal widths? For continuous data with unequal groups, we use histograms where the area of the bar represents the frequency, rather than the height.

The height of the bar in a histogram represents the frequency density, and the vertical axis must be explicitly labelled with this term.

$$\text{Frequency Density} = \text{Frequency} \div \text{Class Width}$$

Worked Example: Finding Frequency Density

Calculate the frequency density for the class interval $10 \le t < 30$ with a frequency of 12.

Step 1: Calculate the class width.

• $\text{Class Width} = 30 - 10 = 20$

Step 2: Substitute into the formula.

• $\text{Frequency Density} = 12 \div 20$

Step 3: Calculate the final answer.

• $\text{Frequency Density} = 0.6$

Sketching and Interpreting Box Plots

You can list a hundred data points, but a box plot summarizes the entire spread of the data using just five numbers. This is known as the five-number summary.

To construct a box plot, you need:

1. Minimum Value
2. Lower quartile ($Q_1$)
3. Median ($Q_2$)
4. Upper quartile ($Q_3$)
5. Maximum Value

Here is an approximate diagram of a box plot showing these key features:

            Lower Quartile                    Upper Quartile
Minimum          (Q1)           Median             (Q3)          Maximum
   |--------------+---------------|-----------------+---------------|
                  |                                 |
                  +---------------------------------+
                                  IQR

A rectangular box is drawn from the lower quartile to the upper quartile, showing the middle 50% of the data. A vertical line is drawn inside the box at the median, and straight "whiskers" extend from the box outwards to the minimum and maximum values.

If you are extracting these values from a cumulative frequency curve with a total frequency of $n$, the median is read at $0.5n$, the lower quartile at $0.25n$, and the upper quartile at $0.75n$. For discrete data, the median position is calculated as $\frac{n+1}{2}$.

Outliers and Skewness (Higher Tier)

Sometimes the highest or lowest values in a dataset do not fit the general pattern and are classified as an outlier. Edexcel commonly uses the "1.5 $\times$ IQR" rule to identify them, where the interquartile range (IQR) is calculated as $Q_3 - Q_1$.

A value is an outlier if it falls outside these boundaries:

• Lower Boundary: $< Q_1 - (1.5 \times \text{IQR})$
• Upper Boundary: $> Q_3 + (1.5 \times \text{IQR})$

On a box plot, outliers are marked with a distinct cross (X). If an outlier is present, the whisker must end at the smallest or largest value in the dataset that is not an outlier, rather than extending to the outlier boundary itself.

The position of the median inside the box also indicates skewness:

• Positive Skew: The right tail is longer, and the median is closer to the lower quartile.
• Negative Skew: The left tail is longer, and the median is closer to the upper quartile.
• Symmetrical: The median is exactly in the center of the box.

Comparing Distributions

When asked to compare two distributions of data, Edexcel mark schemes strictly require exactly two distinct points of comparison. You must interpret these comparisons in the context of the original problem.

Use the following paired structure to ensure you secure full marks:

Comparison Point	Metric to Use	What It Means in Context
Central Tendency (Average)	Compare the Medians. Do not use the mean when comparing box plots.	States which group was higher/lower on average. Example: "The median for Class A (15) is higher than Class B (12), so on average, Class A scored more marks."
Spread (Consistency)	Compare the Interquartile Range (IQR) or the Range.	A smaller IQR/Range means the data is more consistent. Example: "The IQR for Class A (4) is lower than Class B (7), so Class A's marks were more consistent."