Revision and practice with answers
On this page 4 headings
Use this page after the lessons. Attempt each question before opening its answer.
Quick revision
| Idea | Remember |
|---|---|
| Frequency | Count the cases in each category. |
| Relative frequency | Count divided by group total. |
| Percent / pie angle | Share × 100% / share × 360°. |
| Bar / pie / Pareto | Compare categories / parts of whole / descending counts. |
| Ordinal charts | Keep the natural category order. |
| Fair chart | Clear labels, honest areas and zero-baseline bars. |
| Mode / median | Most common / middle category after sorting. |
Practice with explained answers
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1. Count A,B,A,C,B,A,D,C,A,D,A,B.
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A = 5, B = 3, C = 2, D = 2; total = 12.
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2. Find the shares and percentages for question 1.
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A: 5/12 ≈ 41.67%. B: 3/12 = 25%. C and D: each 2/12 ≈ 16.67%. Rounded percentages total 100.01%.
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3. Find the pie angles for question 1.
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A = 150°, B = 90°, C = 60°, D = 60°; total = 360°.
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4. Which category is the mode in question 1?
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A. The mode is the category; 5 is its frequency.
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5. Counts are X = 10, Y = 40, Z = 20. Give Pareto order.
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Y, Z, X, with counts 40,20,10.
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6. Two bars represent 50 and 60 but start at 40. What impression can they create?
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Visible heights are 10 and 20, suggesting a doubling. The true increase is 10/50 = 20%.
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7. Ordered responses are Poor,Poor,Fair,Good,Good,Good,Excellent. Find mode and median.
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Mode = Good, appearing 3 times. Median = the 4th ordered response = Good.
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8. A multiple-choice survey allows more than one favorite sport. Should percentages that total 150% form one pie?
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No. Overlapping responses do not form mutually exclusive parts of one 100% whole.
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9. For ordinal responses S,S,M,L,L,L, what are the middle categories?
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Positions 3 and 4 are M and L. Report both; do not average label codes. Follow the course convention if a single median is requested.
More practice: apply what you learned
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10. Count Tea,Coffee,Tea,Juice,Tea,Coffee,Juice,Tea,Coffee,Tea.
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Tea = 5, Coffee = 3, Juice = 2; total = 10. Check that 5+3+2 = 10.
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11. For question 10, find relative frequencies and percentages.
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Tea: 5/10 = 0.5 = 50%. Coffee: 3/10 = 0.3 = 30%. Juice: 2/10 = 0.2 = 20%. The exact shares sum to 1, and the percentages to 100%.
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12. For question 10, calculate all pie angles.
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Tea: 0.5×360° = 180°. Coffee: 0.3×360° = 108°. Juice: 0.2×360° = 72°. Total = 360°.
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13. For question 10, identify the mode and decide whether it is a majority.
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Mode = Tea. Its share is exactly 50%, so it is half the group, not a strict majority. A majority requires more than 50%.
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14. A category has frequency 7 in 28 responses. Find its share, percentage and pie angle.
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Share = 7/28 = 0.25; percentage = 25%; angle = 0.25×360° = 90°.
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15. A chart says 30% of 50 students chose A. How many students is that?
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0.30×50 = 15 students. Multiplying the proportion by the total recovers the category count.
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16. One group has 18 of 30 using the library; another has 24 of 60. Which has the larger count and which the larger share?
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The second has the larger count (24 versus 18). The first has the larger share: 18/30 = 60%, compared with 24/60 = 40%.
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17. A bar chart uses relative frequency. The heights are 0.2,0.5,0.3. What are the corresponding percentages?
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20%,50%,30%. Multiply each proportion by 100%. The bar heights already sum to 1.
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18. Defect counts are A=3,B=8,C=5,D=1. Give Pareto order and the mode.
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Pareto order: B,C,A,D. Mode = B, with the highest frequency 8.
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19. A pie chart shows shares 45%,35%,30%. Is this a complete, non-overlapping one-response distribution?
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No. The total is 110%, too far from 100% to be ordinary rounding. Check overlapping responses, wrong denominators or a data error.
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20. Three equal shares are displayed as 33.3% each. Explain the total.
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They total 99.9% because each exact share 1/3 was rounded. The original shares still sum to 1. Calculate from exact counts or unrounded shares.
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21. A picture grows from width 2 cm and height 3 cm to width 4 cm and height 6 cm. How much did its area change?
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Original area = 2×3 = 6 cm². New area = 4×6 = 24 cm², four times as large. It would exaggerate data that only doubled.
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22. With order Poor<Fair<Good<Excellent, responses are Poor,Fair,Good,Good,Excellent. Find mode and median.
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Mode = Good, appearing twice. The 5 observations are already ordered; median is position 3 = Good.
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23. With grade order A,B,C,D, data are A,A,A,A,B,B,C. Find mode and median.
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Mode = A (4 times). Median is position 4 of 7 = A. Do not confuse position 4 with grade D.
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24. Ordered responses are Low,Low,Medium,High. What are the middle categories, and can you average their codes?
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Positions 2 and 3 are Low and Medium. Report both middle categories and follow the course convention if one is requested. Averaging arbitrary codes does not create a meaningful halfway category.
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25. Counts are A=4,B=4,C=2. What are the modes?
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A and B tie for the highest frequency, so both are modes: the data are bimodal. Do not choose one arbitrarily.
Ready to move on?
Explain each answer in your own words. If you cannot explain the denominator, category order or measurement scale, revisit that lesson before continuing.