Statistical charts and diagrams
Section: Statistics | Syllabus: Cambridge IGCSE Mathematics (0580)
Stem-and-Leaf Diagrams
- A stem-and-leaf diagram splits each data value into a stem (all digits except the last) and a leaf (the last digit), keeping the original data visible while also showing its overall shape
- In an ordered stem-and-leaf diagram, the leaves in each row are arranged in ascending order
- A key must always be included, e.g. "1 | 1 represents 11", so the place value of each stem-leaf pair is unambiguous
Worked Example: Constructing a Stem-and-Leaf Diagram
- Question: The number of minutes a bus was delayed is recorded on 12 different days: 24,6,31,15,9,27,6,18,11,33,20,14. Construct an ordered stem-and-leaf diagram, using the tens digit as the stem.
- Step 1: Order the data: 6,6,9,11,14,15,18,20,24,27,31,33
- Step 2: Split each value into its stem (tens digit) and leaf (units digit) and group by stem
- Answer:
Key: 1 | 1 represents 11 minutesStem Leaves 0 6 6 9 1 1 4 5 8 2 0 4 7 3 1 3
Common Mistakes
MistakeLeaving the leaves in each row in the order the data was originally collected, instead of ascending order
Fixfor a fully ordered stem-and-leaf diagram, sort the leaves within each row from smallest to largest
MistakeOmitting the key, or writing an unclear one, leaving the diagram open to more than one interpretation
Fixalways include a key that shows exactly what one stem-leaf pair represents - without it, the diagram cannot be read correctly
Pie Charts
- A pie chart shows data as sectors of a circle; since a full circle is 360°, each sector's angle is proportional to its frequency: sector angle=(frequency/total frequency)360°
- To find a frequency from a given sector angle, rearrange the formula: frequency=(angle/360°)
- All the sector angles in a pie chart must add up to exactly 360°
Worked Example: Using Sector Angles in a Pie Chart
- Question: Kondwani asks 60 people their favourite season: summer, winter or autumn, and represents the results in a pie chart. The number who prefer winter is represented by a sector angle of 84°. (a) Calculate the number of people who prefer winter. (b) Three times as many people prefer summer as prefer autumn. Calculate the sector angle representing summer.
- Step 1 (a): (84/360)60=14
- Step 2 (b): Angle remaining for summer and autumn combined =360-84=276°
- Step 3 (b): Summer is three times autumn, so summer =3 parts and autumn =1 part, a total of 4 parts. One part =2764=69°
- Step 4 (b): Summer sector =369=207°
- Answer: (a) 14 people prefer winter; (b) the summer sector angle is 207°
Common Mistakes
MistakeForgetting to divide by 360° when converting a sector angle back into a frequency (or forgetting to multiply by 360° when finding an angle)
Fixa full pie chart is always 360° - this value must appear in both the angle-to-frequency and frequency-to-angle conversions
MistakeSplitting a ratio using the full 360° instead of first subtracting any sector angles that are already known
Fixsubtract every already-known sector angle from 360° first, then split only the remaining angle according to the given ratio
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