Ch 2 · 9.4–9.5 Statistical Charts and Scatter Diagrams · O Level Mathematics 4024
Topic 2 of 3 in Statistics — O Level Mathematics 4024 — 48 lessons.
Introduction
A good chart shows a pattern at a glance: which flavour is most popular, how a budget is split, whether taller people tend to have longer arms. This lesson covers the standard charts and the scatter diagram, which shows a relationship between two variables. (Syllabus sections 9.4 and 9.5.)
Key ideas — Bar charts and pictograms
Bars of equal width with gaps (for categories or discrete data), a labelled frequency axis with an even scale. Dual (side-by-side) bar charts compare two groups; composite (stacked) bars show parts of a total. A pictogram uses a symbol with a key (e.g. one circle = 4 people).
🧑🏫 Teacher's explanation
Imagine you are organizing a school sports day. If you want to show how many students chose football versus tennis, a bar chart is your best friend. Each bar represents a category, and the height of the bar tells you the frequency. Think of it like building towers of blocks; if the football tower is twice as high as the tennis tower, you know twice as many students chose football. Crucially, the width of the bars must be identical so the viewer isn't tricked into thinking a wider bar means 'more'.
📖 Definitions
🔍 Worked example — Easy
A pictogram uses a symbol of a star star to represent 5 students. If there are 3 stars next to 'Maths' and 4 stars next to 'Science', calculate the total number of students in both subjects.
🔍 Worked example — Standard
In a dual bar chart comparing boys and girls in two classes, Class A has 12 boys and 15 girls. Class B has 10 boys and 18 girls. Calculate the total number of girls in both classes.
🔍 Worked example — Exam-style
A composite bar chart shows total sales for a shop. The total bar height for 'Monday' is 50 units. The 'Fruit' segment is 30 units and the 'Veg' segment is 20 units. If the 'Fruit' segment is represented by a blue color and 'Veg' by green, what percentage of Monday's sales was 'Fruit'?
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Pie charts
Each sector angle is proportional to the frequency:
🧑🏫 Teacher's explanation
Imagine you have a giant pizza that represents a whole group of people. If 5 people out of 20 want pepperoni, that is 5/20 of the group. To show this on a pizza, you wouldn't just cut a random slice; you would cut a slice that takes up exactly 5/20 of the total 360^° of the circle.
📖 Definitions
🔍 Worked example — Easy
A class of 40 students was asked their favorite fruit. 10 students chose Apple, 15 chose Banana, and 15 chose Orange. Calculate the angle for the 'Apple' sector.
🔍 Worked example — Standard
In a survey of 200 people, 40 people said they prefer Tea. Calculate the angle for the 'Tea' sector and the percentage of people who prefer Tea.
🔍 Worked example — Exam-style
A pie chart shows three categories: A, B, and C. The angle for A is 120^°, and the angle for B is 150^°. If the total number of items represented by the chart is 120, find the frequency of category C.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Stem-and-leaf diagrams (0580)
The stem shows the leading digit(s) and the leaves the last digit, in order, with a key (e.g. 3 mid 7 means 37). Back-to-back diagrams compare two sets.
Key ideas — Scatter diagrams and correlation
Plot each pair as a small cross. Then describe the pattern:
🧑🏫 Teacher's explanation
Imagine you are looking at a crowd of people. If you notice that as people get taller, their shoe size also gets larger, you have spotted a pattern. In statistics, we use scatter diagrams to visualize these patterns between two different measurements, called variables. Instead of just looking at a list of numbers, we plot them as points on a grid to see if they 'move together'.
📖 Definitions
🔍 Worked example — Easy
A student plots the following (x, y) coordinates: (1, 2), (2, 4), (3, 6), (4, 8). Describe the correlation.
🔍 Worked example — Standard
A scatter diagram shows points that form a downward sloping pattern from left to right. Describe the correlation and the type of relationship.
🔍 Worked example — Exam-style
A scatter diagram is drawn for the following data: (1, 5), (2, 2), (3, 8), (4, 1), (5, 9). Describe the correlation and explain why a line of best fit might be difficult to draw.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
\begin{itemize} \item[\text{Data:}] (1, 20), (2, 35), (3, 30), (4, 50), (5, 45), (6, 65), (7, 60), (8, 75), (9, 70), (10, 85) \end{itemize} (a) Describe the correlation shown in the data. (b) Draw a line of best fit for this data. (c) Use your line of best fit to estimate the score for a student who studied for 5.5 hours. [4 marks]
Key ideas — Line of best fit
Drawn by eye as a single ruled line through the middle of the points, across the full data set, with roughly equal numbers of points on each side (it need not pass through any point or the origin). Use it to estimate a value: reading within the range of the data is reliable; going beyond it (extrapolation) is not.
🧑🏫 Teacher's explanation
Imagine you are tracking the height of a growing plant every week. The measurements might not fall in a perfectly straight line because sometimes the plant grows a lot, and sometimes it grows a little. However, if you look at the overall trend, you can see a general direction. A line of best fit is like a 'trendline' that cuts through the middle of all your measurements to show that general direction.
📖 Definitions
🔍 Worked example — Easy
A scatter diagram shows the relationship between hours studied (x) and test scores (y). The points suggest a positive correlation. If the line of best fit passes through (2, 40) and (6, 80), find the estimated score for 3 hours of study.
🔍 Worked example — Standard
A scatter diagram shows the relationship between temperature (T) and ice cream sales (S). The line of best fit is S = 5T + 10. Estimate the sales when the temperature is 25^° C.
🔍 Worked example — Exam-style
A scatter diagram shows the weight of packages (w) and the cost of shipping (c). The line of best fit is drawn through the data. A student uses the line to estimate the cost for a package weighing 50kg, but the data only goes up to 30kg. Explain why this estimate might be unreliable.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) Describe the correlation shown. [1] (b) A line of best fit is drawn through the points. Use the line to estimate the number of mistakes for a class of 25 students. [1] [2 marks]
Worked examples
The siblings data (4, 9, 8, 5, 2 out of 28) give angles 4/28 × 360 = 51.4^°, then 115.7^°, 102.9^°, 64.3^° and 25.7^° (they add to 360^°).
🔍 Worked example — Standard
A scatter diagram shows the relationship between hours of revision (x) and test marks (y). A line of best fit is drawn through the points (2, 40) and (8, 82). Find the estimated mark for a student who revised for 6 hours.
🔍 Worked example — Exam-style
A pie chart represents 28 siblings. The frequencies for different categories are 4, 9, 8, 5, and 2. Calculate the angle for the category with frequency 4.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 25/60 × 360 = 150^°. (2) 3tfrac12 symbols. (3) negative correlation. (4) the relationship may not continue outside the range of the data (extrapolation is unreliable).