Introduction to Cumulative Frequency and Box Plots

Welcome! In this chapter, we are going to learn how to take large sets of data and turn them into clear, visual stories. While a simple list of numbers can be overwhelming, Cumulative Frequency Charts and Box Plots allow us to see the "shape" of the data instantly. We will learn how to find the middle values, see how spread out the data is, and spot any unusual results called outliers.

These tools are essential in the "Processing, representing and analysing data" stage of the Statistical Enquiry Cycle. Whether you are looking at exam results, heights of plants, or race times, these diagrams help you make sense of the "big picture."

Note: This chapter builds on your knowledge of averages and ranges. If you need a quick reminder: the median is the middle value, and the range is the difference between the highest and lowest values.

1. Cumulative Frequency

The word "cumulative" simply means "adding up as you go." Think of it like a "running total" in a video game score or a bank balance.

Creating a Cumulative Frequency Table

Before we can draw a chart, we need to adjust our frequency table. We add a new column for Cumulative Frequency. To find each value, add the frequency of that group to the sum of all the frequencies before it.

Example: If you have test scores in groups:

  • Scores \(0 < s \le 10\): Frequency = \(5\) → Cumulative Frequency = \(5\)
  • Scores \(10 < s \le 20\): Frequency = \(8\) → Cumulative Frequency = \(5 + 8 = 13\)
  • Scores \(20 < s \le 30\): Frequency = \(2\) → Cumulative Frequency = \(13 + 2 = 15\)

Drawing the Cumulative Frequency Chart

When plotting your points, follow these three golden rules:

  1. The X-axis: Always plot the point at the upper class boundary (the end of the interval). If the group is \(10 < s \le 20\), you plot your point at \(20\).
  2. The Y-axis: Plot the Cumulative Frequency value.
  3. The Shape: Join the points with a smooth, S-shaped curve. It should start at the lowest possible value on the x-axis (where the frequency is \(0\)).

Quick Review: Why the upper boundary? Because by the time we reach the end of the "\(10 \text{ to } 20\)" group, we have accounted for all the people who scored up to \(20\).

2. Finding Key Values from the Chart

Once your curve is drawn, you can find the "quartiles." Quartiles split your data into four equal quarters.

  • The Median (\(Q_2\)): This is the middle value. Find \(n/2\) (half of the total frequency) on the y-axis, move across to the curve, and read down to the x-axis.
  • Lower Quartile (\(LQ\) or \(Q_1\)): This is one-quarter of the way through the data. Find \(n/4\) on the y-axis and read the value on the x-axis.
  • Upper Quartile (\(UQ\) or \(Q_3\)): This is three-quarters of the way through the data. Find \(3n/4\) on the y-axis and read the value on the x-axis.
  • Interquartile Range (\(IQR\)): This measures the spread of the middle \(50\%\) of the data.
    Formula: \(IQR = UQ - LQ\)

Did you know? The \(IQR\) is often more reliable than the standard range because it ignores extreme values (outliers) at the very top or bottom.

3. Box Plots (Box and Whisker Diagrams)

A Box Plot is a 1D summary of your cumulative frequency curve. It uses five key pieces of information, known as the five-number summary:

  1. Lowest Value (the start of the "whisker")
  2. Lower Quartile (the start of the "box")
  3. Median (the line inside the box)
  4. Upper Quartile (the end of the box)
  5. Highest Value (the end of the other "whisker")

Common Mistake: Students often forget to draw the median line inside the box. Without it, the box plot doesn't show where the center of the data is!

4. Identifying Outliers

An outlier is a piece of data that doesn't seem to fit the rest of the pattern—it's either much higher or much lower than the other values.

Identifying by Inspection (Foundation and Higher)

You can often spot an outlier just by looking at a list of data or a diagram. For example, if most students score between \(40\) and \(70\) marks, but one student scores \(2\), that \(2\) is an outlier.

Identifying by Calculation (Higher Tier Only)

In the Higher Tier, you must use specific "fences" to prove a value is an outlier. A value is an outlier if it is:

  • Smaller than \(LQ - (1.5 \times IQR)\)
  • Larger than \(UQ + (1.5 \times IQR)\)

Note: For Higher Tier, outliers can also be defined as being more than \(3\) standard deviations away from the mean, but for box plots, the \(1.5 \times IQR\) rule is the standard method.

5. Skewness

Skewness describes how "lopsided" the distribution is. You can determine this by looking at your box plot or cumulative frequency curve.

  • Symmetrical: The median is roughly in the middle of the box, and the whiskers are of similar length.
  • Positive Skew: The "tail" of the data is on the right (higher values). In a box plot, the median is closer to the Lower Quartile.
  • Negative Skew: The "tail" of the data is on the left (lower values). In a box plot, the median is closer to the Upper Quartile.

Memory Trick: Think of where the "tail" is. If the long tail points toward the positive (larger) numbers on the number line, it's Positive Skew.

6. Comparing Data Sets

One of the most common exam questions asks you to compare two sets of data using their box plots. When you do this, you must comment on two things:

  1. The Average: Compare the Medians. (e.g., "Class A had a higher median score than Class B, meaning they generally performed better.")
  2. The Spread: Compare the Interquartile Ranges (\(IQR\)). (e.g., "Class B had a smaller \(IQR\) than Class A, meaning their scores were more consistent.")

Important: Always use the context of the question (e.g., talk about "test scores" or "heights," not just "the numbers").

Chapter Summary

  • Cumulative Frequency is a running total, plotted at upper class boundaries.
  • The Median is the middle value; Quartiles split the data into quarters.
  • The IQR (\(UQ - LQ\)) shows the spread of the middle \(50\%\) of the data.
  • Box Plots visually represent the five-number summary.
  • Higher Tier: Outliers are calculated using \(1.5 \times IQR\) beyond the quartiles.
  • When comparing, always mention the Median (average) and the IQR (consistency).