Statistical Charts and Diagrams (IGCSE 0580 Mathematics)
Hello there, future statistician! Welcome to the world of visualizing data. Numbers are great, but sometimes they look like a boring list. This chapter is all about turning those dull lists into beautiful, informative pictures! These pictures—charts and diagrams—help us quickly interpret data and make smart conclusions. Don't worry if you aren't an artist; we just need to know how to draw and read them accurately!
Understanding Data Types: Discrete vs. Continuous
Before we draw a chart, we must know what kind of data we have. Remember these definitions from earlier in the Statistics chapter (C10.3 / E10.3):
- Discrete Data: Data that can only take specific, fixed values (usually whole numbers) and are often counted.
Example: The number of siblings a student has (you can't have 2.5 siblings). - Continuous Data: Data that can take any value within a range, and are usually measured.
Example: Height, weight, or temperature. (Note: Most charts in C10.6 deal with discrete or categorical data.)
Quick Review Box: Discrete data uses separate bars (like a standard bar chart). Continuous data often requires methods like histograms (which are generally covered in the Extended syllabus for 0580, but are outside the scope of C10.6/E10.6 provided here).
1. Bar Charts and Diagrams
Bar charts are one of the most common ways to represent data. They use rectangular bars, where the length (or height) of the bar is proportional to the frequency (how often something occurs).
Drawing a Simple Bar Chart (C10.6/E10.6(a))
Key Features:
- The bars are usually drawn vertically (up and down).
- There must be equal gaps between the bars.
- The axes must be clearly labelled and have a suitable, consistent scale.
Example: Showing the favourite sport of 50 students.
The Sport category (Football, Rugby, Tennis) goes on the horizontal axis (x-axis), and the Frequency (number of students) goes on the vertical axis (y-axis).
Composite (Stacked) Bar Charts
A composite bar chart displays different components *stacked* on top of each other within a single bar. It shows both the total frequency and the proportion of each sub-category.
Think of it like a Layer Cake: The whole bar is the total number of people in a category, and the different layers are the smaller groups within that category.
Important: You must include a key (legend) to show what each colour or pattern in the stack represents.
Dual (Side-by-Side) Bar Charts
A dual bar chart is used when you need to compare two different data sets for the same categories side-by-side.
Example: Comparing the sales figures of "Product A" and "Product B" across four different months.
For each month, you would have two bars standing right next to each other (one for A, one for B).
Important: Like the composite chart, a clear key is essential for telling the two data sets apart.
Key Takeaway for Bar Charts: Bars must have gaps, scales must be consistent, and if you have complex bars (stacked/dual), you need a key!
2. Pie Charts
Pie charts (C10.6/E10.6(b)) are circular charts used to show how a whole quantity is divided into parts. The total circle represents 100% of the data, or 360°.
Think of it like a Pizza: The whole pizza is your total data set, and each slice represents a category.
Calculating Angles for a Pie Chart (The Key Skill!)
The size of each sector (slice) in a pie chart is proportional to the frequency it represents. Since a full circle is 360°, we calculate the angle for each category using the total frequency.
Step-by-Step Calculation:
- Find the Total Frequency (sum up all the numbers).
- Calculate the Angle for each category using the formula:
$$ \text{Angle} = \frac{\text{Frequency of Category}}{\text{Total Frequency}} \times 360^{\circ} $$
Example: If 10 students out of a total of 50 students chose "Red" as their favourite colour:
\( \text{Angle for Red} = \frac{10}{50} \times 360^{\circ} \)
\( \text{Angle for Red} = 0.2 \times 360^{\circ} = 72^{\circ} \)
Common Mistake to Avoid: Always ensure that your final calculated angles add up to 360°. If they don't, you made a calculation error!
Key Takeaway for Pie Charts: Pie charts show proportions. You must calculate the angle for each category based on its share of the total 360 degrees.
3. Pictograms
A pictogram (C10.6/E10.6(c)) uses simple images or symbols to represent frequencies.
The Golden Rule: The Key!
Every pictogram MUST have a clear key that explains what one symbol represents.
Example: If a pictogram shows the number of apples sold, the key might state:
Key: \( \text{One apple symbol} = 10 \text{ apples sold} \)
If the frequency is 35, you would draw 3 full apple symbols and one half-apple symbol. Make sure your symbols are easy to draw and interpret fractions of.
Did you know? Pictograms are great for making data easily understandable by a wide audience, even those who might find charts intimidating!
Key Takeaway for Pictograms: Always include a clear key that defines the value of the symbol. Be ready to draw half or quarter symbols if needed.
4. Stem-and-Leaf Diagrams
Stem-and-leaf diagrams (C10.6/E10.6(d)) are a fantastic way to organise a set of numerical data because they preserve the original raw data while showing the shape of the distribution.
Imagine you collected the test scores of 20 students:
45, 61, 68, 49, 70, 72, 83, 61, 55, 59, 78, 81, 74, 60, 52, 53, 67, 70, 60, 75
Drawing a Stem-and-Leaf Diagram (Step-by-Step)
- Identify the Stem and the Leaf:
The Stem is usually the first digit(s) (e.g., the tens or hundreds).
The Leaf is usually the last digit (e.g., the units digit).
For the scores above, the stem will be the tens digit (4, 5, 6, 7, 8) and the leaf will be the units digit. - Draw the Draft Diagram (Unordered):
4 | 5, 9
5 | 5, 9, 2, 3
6 | 1, 8, 1, 0, 7, 0
7 | 0, 2, 8, 4, 0, 5
8 | 3, 1 - Order the Leaves (Crucial Step!): The leaves in each row must be placed in ascending order (smallest to largest).
4 | 5, 9
5 | 2, 3, 5, 9
6 | 0, 0, 1, 1, 7, 8
7 | 0, 0, 2, 4, 5, 8
8 | 1, 3 - Include a Key (MANDATORY): The key tells the reader how to interpret the numbers.
Key: 5 | 2 means 52 (marks)
Accessibility Tip: The order of the leaves is critical! If you forget to order the leaves, your diagram is incorrect. Always put the key next to the diagram.
Interpreting the Stem-and-Leaf Diagram:
This diagram clearly shows the distribution. You can see that most students scored in the 60s and 70s. You can also easily find the mode (most common number) and the median (middle number) from the ordered list.
Key Takeaway for Stem-and-Leaf Diagrams: They show the data's shape while keeping the original values. They must be ordered and have a key.
5. Simple Frequency Distributions
Simple frequency distributions (C10.6/E10.6(e)) are usually the tables we create before drawing any chart, often including a tally. They are the simplest way to summarize data.
Tally Tables and Frequency Tables (C10.1):
This involves counting how many times each value or category occurs. This raw count is the frequency.
| Category | Tally | Frequency |
| --- | --- | --- |
| Blue | IIIII I | 6 |
| Green | IIII | 4 |
We use these tables to organize the data, which then allows us to calculate measures of average (mean, mode, median) or draw the charts discussed above (Bar charts, Pie charts, etc.).
Key Takeaway: Simple frequency distributions are the backbone of all statistical charts. They list the categories and the counts (frequencies) for each.
Final Review: Interpreting and Comparing Data (C10.2 / E10.2)
The whole point of charts is to interpret the data and draw inferences. In the exam, you will often be asked to compare two charts or sets of data.
When comparing data sets, look at two main things:
- Averages (Central Tendency): Where is the middle of the data?
Example: "Set A has a higher mean, so students in Set A performed better overall." - Spread (Range/Interquartile Range): How consistent or spread out is the data?
Example: "Set B has a smaller range, meaning their scores were more consistent."
Important Reminder (C10.2.3): You must always appreciate the restrictions on drawing conclusions. A small sample size or a biased selection process means your conclusions might not be reliable for the entire population.
Keep practicing drawing these charts neatly and accurately—especially the pie chart calculations and the ordered stem-and-leaf diagrams. You've got this!