Introduction to Presenting Data

Data by itself can be a messy pile of numbers. To make sense of it, we need to organize and visualize it. In this chapter, we will learn how to turn raw information into frequency tables, histograms, and box plots. These tools help us "see" the shape of the data, find the middle (central tendency), and see how spread out the values are (dispersion). Whether you are analyzing test scores or the heights of athletes, these techniques are the foundation of all statistics.

1. Frequency Tables

A frequency table is simply a way to organize data by counting how often each value (or range of values) occurs.

Discrete vs. Grouped Data

  • Discrete Data: Used for things you can count (like the number of pets). We list each value and its frequency.
  • Grouped Data (Continuous): Used for things you measure (like time or weight). Because there are infinite possible values, we group them into class intervals.

Example of Grouped Data:
If we measure the heights of students, we might use intervals like \(150 \le h < 160\).
Common Mistake: Make sure the intervals don't overlap! The notation \(150 \le h < 160\) means a student who is exactly \(160\) cm tall goes into the next group.

Key Takeaway: Frequency tables reduce the "noise" of raw data, making it easier to calculate the mean or identify the modal class (the group with the highest frequency).

2. Histograms

A histogram looks like a bar chart, but there is a key difference: the bars touch each other because they represent continuous data.

How to Draw a Histogram

  1. The horizontal axis (x-axis) shows the variable being measured (the class intervals).
  2. The vertical axis (y-axis) shows the frequency.
  3. The width of each bar represents the class width.
  4. The bars must be adjacent (no gaps) unless a frequency is zero.

Did you know? In IB Mathematics: AI, we usually deal with histograms that have equal class widths. This means the height of the bar tells you exactly how many items are in that group.

Quick Review: If you see a gap between bars, it’s a Bar Chart (for categorical data). If the bars touch, it’s a Histogram (for continuous data).

3. Cumulative Frequency Graphs

Cumulative frequency is a "running total" of frequencies. It tells us how many data points fall below a certain value.

Step-by-Step: The S-Curve

  1. Add a "Cumulative Frequency" column to your table by adding each frequency to the sum of the previous ones.
  2. Plot the points using the upper boundary of each class interval on the x-axis and the cumulative frequency on the y-axis.
  3. Connect the points with a smooth curve (often called an ogive). It should look like a stretched-out "S".

Using the Graph to Find Key Values

This graph is a powerful tool for finding percentiles:

  • Median (\(Q_2\)): Go to \(50\%\) of the total frequency on the y-axis, move across to the curve, and read the value on the x-axis.
  • Lower Quartile (\(Q_1\)): Go to \(25\%\) of the total frequency.
  • Upper Quartile (\(Q_3\)): Go to \(75\%\) of the total frequency.
  • Interquartile Range (IQR): Calculated as \(IQR = Q_3 - Q_1\). This tells you the range of the middle \(50\%\) of the data.

Key Takeaway: Cumulative frequency graphs help us see the distribution of data and pinpoint exactly where a certain percentage of the population lies.

4. Box and Whisker Diagrams

A box and whisker diagram (or box plot) is a visual summary of the "five-number summary" of a data set. It’s a great way to compare two different sets of data side-by-side.

The Five-Number Summary

  1. Minimum value: The smallest data point (that isn't an outlier).
  2. Lower Quartile (\(Q_1\)): The start of the box.
  3. Median (\(Q_2\)): The line inside the box.
  4. Upper Quartile (\(Q_3\)): The end of the box.
  5. Maximum value: The largest data point (that isn't an outlier).

Analogy: Think of the "box" as the heart of the data. Half of all your data points live inside that box!

5. Outliers

Sometimes data contains "weird" values that are much higher or lower than the rest. In the IB curriculum, we have a specific rule to decide if a value is an outlier.

The \(1.5 \times IQR\) Rule

A value is an outlier if:

  • It is smaller than \(Q_1 - 1.5 \times IQR\)
  • It is larger than \(Q_3 + 1.5 \times IQR\)

Important Tip: On a box plot, outliers are usually marked with an "x" or a dot. The "whiskers" of the plot then only extend to the smallest and largest values that are not outliers.

Example: If \(Q_1 = 10\), \(Q_3 = 20\), then \(IQR = 10\).
Upper boundary for outliers = \(20 + (1.5 \times 10) = 35\).
Lower boundary for outliers = \(10 - (1.5 \times 10) = -5\).
Any value above \(35\) or below \(-5\) is an outlier.

Summary Checklist

Before you finish this chapter, make sure you can:

  • Construct a frequency table from raw data.
  • Draw a histogram and label the axes correctly.
  • Plot a cumulative frequency curve and use it to find the median and quartiles.
  • Sketch a box and whisker plot using the five-number summary.
  • Identify outliers using the \(1.5 \times IQR\) formula.

Don't worry if reading the graphs feels a bit slow at first. With your GDC (Graphic Display Calculator) and a little practice, identifying these values will become second nature! For more on how to calculate the mean and standard deviation, see the chapter on "Central tendency and dispersion."