Welcome to the World of Data!

Have you ever looked at a massive list of numbers and felt your head spin? You aren't alone! In this chapter, Presentation of Data, we learn how to turn those messy lists of numbers into beautiful, clear pictures. These pictures (or charts) help us spot patterns, make decisions, and tell stories with facts. Whether you are looking at your exam marks or checking the weather, data presentation is everywhere!

1. Stem-and-Leaf Diagrams

Imagine you have a list of test scores: \( 52, 55, 61, 63, 63, 70, 78 \). A Stem-and-leaf diagram is a clever way to organize these numbers so you can still see every single original value while seeing the "shape" of the group.

How it works:

We split each number into two parts: the Stem (the tens digit) and the Leaf (the units digit).

For the score \( 52 \):
Stem: \( 5 \)
Leaf: \( 2 \)

Important Rules:

  • The Key: You must always include a key. For example, Key: \( 5 | 2 \) means \( 52 \) marks.
  • Order: Leaves must be arranged in ascending order (from smallest to largest).
  • Alignment: Keep your numbers in straight columns so it looks like a bar chart on its side!

Quick Tip: If you see the same number twice (like two students scoring \( 63 \)), you must write the leaf \( 3 \) twice!

Key Takeaway: Stem-and-leaf diagrams are great for small data sets because they don't "hide" any of the original numbers.

2. Histograms

A Histogram looks a bit like a bar chart, but it is used for continuous data (things we measure, like height, weight, or time) that has been grouped into intervals.

Key Features:

  • No Gaps: Unlike bar charts, there are no gaps between the vertical bars. This shows that the data is continuous.
  • X-axis: Represents the class boundaries (the measurements).
  • Y-axis: Represents the frequency (how many items fall into that group).

Example: If we are measuring the heights of students in classes of \( 150 \le h < 160 \) cm, the bar will sit exactly between the marks for \( 150 \) and \( 160 \) on the horizontal axis.

3. Frequency Polygons and Curves

If you want to see the "trend" or the "flow" of data more clearly than a histogram, we use Frequency Polygons.

How to construct a Frequency Polygon:

  1. Find the class mark (the middle point) of each group in your frequency table. For a group \( 10 - 20 \), the class mark is \( \frac{10+20}{2} = 15 \).
  2. Plot a point at the class mark (x-coordinate) and the frequency (y-coordinate).
  3. Connect the points with straight lines.
  4. Closing the loop: To make it a "polygon," connect the ends to the horizontal axis at the class marks of the imaginary "empty" groups before and after your data.

What is a Frequency Curve?
It is exactly like a polygon, but instead of using a ruler to draw straight lines, you draw a smooth, free-hand curve through the points. This is used when we have a very large amount of data.

4. Cumulative Frequency: The "Running Total"

Cumulative Frequency is just a fancy way of saying "running total." It tells us "how many items are less than or equal to this value."

Cumulative Frequency Polygons and Curves

To draw this, we plot the cumulative frequency against the upper class boundaries. The graph always goes up (it never goes down because a running total can't decrease!). It usually looks like a stretched-out "S" shape.

What can we find from the Curve?

This is the most powerful tool in this chapter! You can find:

  • Median: The middle value. Find the \( 50\% \) mark on the vertical axis, go across to the curve, and then down to the horizontal axis.
  • Quartiles:
    - Lower Quartile (\( Q_1 \)): The \( 25\% \) mark.
    - Upper Quartile (\( Q_3 \)): The \( 75\% \) mark.
  • Percentiles: Any percentage you want! For example, the \( 90^{th} \) percentile is the value below which \( 90\% \) of the data falls.

Don't worry if this seems tricky at first! Just remember: Vertical axis = Percent/Rank, Horizontal axis = The actual measurement.

5. Choosing the Right Chart

Not every chart is right for every job. Here is a quick guide:

  • Bar Charts: Best for comparing different categories (e.g., favorite fruits).
  • Pie Charts: Best for showing parts of a whole (e.g., how you spend your 24-hour day).
  • Broken Line Graphs: Best for showing how something changes over time (e.g., stock prices or temperature during the day).
  • Histograms: Best for showing the distribution of measured, grouped data.

Did you know? Sometimes you will see Dual Charts. For example, a chart might show rainfall as bars and temperature as a line graph on the same picture. This helps us see if there is a connection between the two!

6. Uses and Abuses of Statistical Charts

Charts can be used to tell the truth, but they can also be used to "lie" or mislead people! Always be a detective when looking at a graph.

Common "Tricks" to watch out for:

  1. The Broken Axis: If the vertical axis doesn't start at \( 0 \), a small difference can look huge! Check the numbers on the side carefully.
  2. Inconsistent Scales: If the gaps between \( 10, 20, 30 \) aren't equal in size, the graph is misleading.
  3. Area vs. Height: In pictograms, if an author doubles the height of a picture but also doubles the width, the area becomes four times bigger, making the increase look much larger than it really is.

Key Takeaway: Always check the scales and the starting points of any chart before you believe what it is telling you!

Quick Review: Use a Stem-and-leaf to keep all data points. Use a Histogram for continuous groups. Use Cumulative Frequency to find the median and quartiles. Be careful of misleading scales!