Reading maps and data — Academy course
120 lessons on reading maps and data across four levels. Practise with educational examples, check your answers and save progress on your device.
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What this page does not do: predict future lottery outcomes.
Course: 120 lessons
- 1. What a marker means
- 2. Online wins
- 3. Amounts and scale
- 4. Comparing counts
- 5. Date and game filters
- 6. Duplicate events
- 7. Missing locations
- 8. Sources and coverage
- 9. Regional share
- 10. Location coverage
- 11. Missing-location share
- 12. Total amount and units
- 13. Events versus markers
- 14. Source verification workload
- 15. Location precision
- 16. Centimetres to metres
- 17. Kilometres from scale
- 18. Length on paper
- 19. Scale denominator
- 20. Enlarged print
- 21. Metres from kilometres
- 22. Kilometres from metres
- 23. Route legs
- 24. Route excess
- 25. Travel time
- 26. East–west difference
- 27. North–south difference
- 28. Rectangle diagonal
- 29. Midpoint: x
- 30. Grid route
- 31. Average prize
- 32. The comparison denominator
- 33. Change in event count
- 34. Percentage change
- 35. Comparison per 1,000 tickets
- 36. Median of selected prizes
- 37. Identifiers and duplicate rows
- 38. Mean monthly count
- 39. Count after exclusion
- 40. Publication delay
- 41. Region percentage
- 42. Count from percentage
- 43. Other regions
- 44. Two disjoint parts
- 45. Percentage-point change
- 46. Events per km²
- 47. Recover the count
- 48. Recover the area
- 49. Area conversion
- 50. Merged areas
- 51. Per thousand tickets
- 52. Per hundred thousand residents
- 53. Rate ratio
- 54. Rate difference
- 55. Pooled rate
- 56. Absolute increase
- 57. Percentage increase
- 58. Percentage decrease
- 59. Base index
- 60. Equal periods
- 61. Largest-prize share
- 62. Periods of different lengths
- 63. Intersection of filters
- 64. Union of filters
- 65. Selection versus full registry
- 66. Publication delay
- 67. Inclusive interval
- 68. Daily average
- 69. Sum of windows
- 70. Overlapping windows
- 71. Mappable count
- 72. Missing percentage
- 73. Coordinate coverage
- 74. Completed locations
- 75. Coverage improvement
- 76. Union of two sources
- 77. Only in first
- 78. Redundant rows
- 79. Primary-source coverage
- 80. Consistent field pairs
- 81. Mean coordinate
- 82. Weighted x
- 83. Weighted y
- 84. Translation effect
- 85. Zero weight
- 86. Class width
- 87. Class number
- 88. Interval count
- 89. Upper endpoint
- 90. Equal counts
- 91. Fourfold count
- 92. Ninefold count
- 93. Double radius
- 94. Half the radius
- 95. Recovering the count
- 96. Leading share
- 97. Top-two share
- 98. Equal partition
- 99. Unequal partition
- 100. After merging boundaries
- 101. Minimum count
- 102. Maximum count
- 103. Minimum share
- 104. Maximum share
- 105. Width of bounds
- 106. First-group rate
- 107. Second-group rate
- 108. Overall rate
- 109. Unweighted average
- 110. Incorrect-weighting difference
- 111. One-axis error
- 112. Two-axis error
- 113. Interval span
- 114. Interval midpoint
- 115. Relative error
- 116. Clean dataset
- 117. Presentation dataset
- 118. Project filter
- 119. Source-completeness check
- 120. Legend-total check
Enter a fraction such as 1/3 or round as instructed beside the question (three or eight decimal places).
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1. What a marker means
Level 1
A marker represents a published event, not a winner’s home. If only a county is published, a reference point is placed near its seat.
Worked example
Two prizes in one town are two events. Exercise counts are fictional teaching data, not actual results.
Watch out
Do not interpret a reference point as the purchase address.
2. Online wins
Level 1
An online win without a published location belongs in the table, not an arbitrary city. No marker does not mean no win.
Worked example
In an example dataset, 2 online wins out of 20 records are 10%.
Watch out
Do not infer winners’ private locations.
3. Amounts and scale
Level 1
Compare amounts in the same currency and unit. A million is 1,000,000. The table shows the amount reported by the source.
Worked example
PLN 3.5 million = PLN 3,500,000.
Watch out
Gross prizes are not take-home amounts. This section does not calculate tax.
4. Comparing counts
Level 1
A count difference describes events in this dataset. It does not automatically describe the chance per ticket.
Worked example
Example: 8 records in A and 3 in B differ by 5.
Watch out
More tickets sold can yield more wins without changing the chance per ticket.
5. Date and game filters
Level 1
A filter changes the visible dataset. Count, table and map must use the same conditions. The date is the draw date reported in the source.
Worked example
Selecting 2020 must exclude a 2021 record.
Watch out
An article’s publication date may be later than the draw date.
6. Duplicate events
Level 1
The same announcement can appear on several sites. Count each win once using game, date and event identity.
Worked example
Two articles about one win do not create two winners.
Watch out
Equal amounts alone do not prove duplication; separate prizes can match.
7. Missing locations
Level 1
Separate record count from marker count. A record may lack coordinates, and several records may share a location.
Worked example
7 records, 2 without a location: 5 can be located.
Watch out
A blank area is not evidence that nobody ever won there.
8. Sources and coverage
Level 1
Check each record’s source, date and location precision. A selection of documented examples is not a complete register.
Worked example
Verifying 4 of 20 records gives 20% verification coverage.
Watch out
Do not use an incomplete map to rank lucky cities or forecast wins.
9. Regional share
Level 1
Divide the region count by the whole selected dataset.
Worked example
4 of 20 example records is 20%.
Watch out
This is dataset share, not ticket odds.
10. Location coverage
Level 1
The share of located records measures map coverage.
Worked example
15 of 20 records are located: coverage 75%.
Watch out
Point precision needs a separate check.
11. Missing-location share
Level 1
Count missing locations as a separate category.
Worked example
4 missing locations in 20 records give 20%.
Watch out
Do not remove missing records from the denominator without explanation.
12. Total amount and units
Level 1
Convert amounts to one unit before adding.
Worked example
2 million and 500 thousand total 2.5 million.
Watch out
Do not add numbers with different units directly.
13. Events versus markers
Level 1
Several events at one location may overlap on the map.
Worked example
5 events in A and 2 in B: 7 events, 2 places.
Watch out
Visible pin count need not equal event count.
14. Source verification workload
Level 1
Remaining checks equal total records minus verified records.
Worked example
30 records, 18 checked: 12 remain.
Watch out
A repeated publication is not independent confirmation.
15. Location precision
Level 1
Separate town-level points from county reference points.
Worked example
Of 20 records, 5 name only the county: 25%.
Watch out
Do not add precision absent from the source.
16. Centimetres to metres
Level 1 · Map scale
A scale of 1:s means s of the same units on the ground. Convert lengths to matching units first.
Worked example
Teaching example. 1:10000; 4 cm on the map. Ground distance in m? 4 × 10000 / 100 = 400
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
17. Kilometres from scale
Level 1 · Map scale
A scale of 1:s means s of the same units on the ground. Convert lengths to matching units first.
Worked example
Teaching example. 1:50000; 3 cm. How many km? 3 × 50000 / 100000 = 1.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
18. Length on paper
Level 1 · Map scale
A scale of 1:s means s of the same units on the ground. Convert lengths to matching units first.
Worked example
Teaching example. 4 km at 1:100000. How many cm? 4 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
19. Scale denominator
Level 1 · Map scale
A scale of 1:s means s of the same units on the ground. Convert lengths to matching units first.
Worked example
Teaching example. 3 cm represents 3 km. Find s in 1:s. 3 × 100000 / 3 = 100000
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
20. Enlarged print
Level 1 · Map scale
A scale of 1:s means s of the same units on the ground. Convert lengths to matching units first.
Worked example
Teaching example. A 1:50000 map is printed twice as large linearly. New denominator? 50000 / 2 = 25000
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
21. Metres from kilometres
Level 1 · Distance units
Convert metres and kilometres before combining them. Road routes can exceed straight-line distance.
Worked example
Teaching example. 4 km: how many m? 4 × 1000 = 4000
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
22. Kilometres from metres
Level 1 · Distance units
Convert metres and kilometres before combining them. Road routes can exceed straight-line distance.
Worked example
Teaching example. 750 m: how many km? 750 / 1000 = 0.75
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
23. Route legs
Level 1 · Distance units
Convert metres and kilometres before combining them. Road routes can exceed straight-line distance.
Worked example
Teaching example. 4 km and 300 m. Total km? 4 + 3/10 = 4.3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
24. Route excess
Level 1 · Distance units
Convert metres and kilometres before combining them. Road routes can exceed straight-line distance.
Worked example
Teaching example. Straight 4 km, road 7 km. Excess km? 7 − 4 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
25. Travel time
Level 1 · Distance units
Convert metres and kilometres before combining them. Road routes can exceed straight-line distance.
Worked example
Teaching example. 120 km at constant 60 km/h. Minutes? 120 / 60 × 60 = 120
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
26. East–west difference
Level 1 · Coordinate grid
These exercises use a flat kilometre grid, not latitude and longitude degrees. Diagonal distances use Pythagoras.
Worked example
Teaching example. Points (4,0) and (7,0). Distance km? 7 − 4 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
27. North–south difference
Level 1 · Coordinate grid
These exercises use a flat kilometre grid, not latitude and longitude degrees. Diagonal distances use Pythagoras.
Worked example
Teaching example. Points (0,4) and (0,9). Distance km? 9 − 4 = 5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
28. Rectangle diagonal
Level 1 · Coordinate grid
These exercises use a flat kilometre grid, not latitude and longitude degrees. Diagonal distances use Pythagoras.
Worked example
Teaching example. Offsets 3 km and 4 km. Straight distance km? √(3² + 4²) = 5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
29. Midpoint: x
Level 1 · Coordinate grid
These exercises use a flat kilometre grid, not latitude and longitude degrees. Diagonal distances use Pythagoras.
Worked example
Teaching example. Endpoint x values: 4 and 10. Midpoint x? (4 + 10) / 2 = 7
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
30. Grid route
Level 1 · Coordinate grid
These exercises use a flat kilometre grid, not latitude and longitude degrees. Diagonal distances use Pythagoras.
Worked example
Teaching example. No diagonals: 4 km east, 3 north. Length km? 4 + 3 = 7
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
31. Average prize
Level 2
The mean of published records describes only those records. A selection of large prizes does not represent all tickets.
Worked example
Example prizes of 2 and 8 million average 5 million.
Watch out
Do not call this the expected payout of a purchased ticket.
32. The comparison denominator
Level 2
Comparing win rates requires ticket counts and matching coverage. Map counts do not provide that denominator.
Worked example
Example: 5 wins per 500 tickets equals 1 per 100. This is arithmetic practice, not game data.
Watch out
Population is not a substitute for tickets sold.
33. Change in event count
Level 2
Absolute growth is the new count minus the previous count.
Worked example
From 7 to 10: an increase of 3.
Watch out
A larger dataset may reflect better collection.
34. Percentage change
Level 2
For a nonzero baseline: (new−old)/old ×100%.
Worked example
From 8 to 10: growth of 25%.
Watch out
This ratio is undefined for a zero baseline.
35. Comparison per 1,000 tickets
Level 2
Use matching wins and ticket counts; scale their ratio to 1,000.
Worked example
2 events per 500 tickets: 4 per 1,000.
Watch out
This is fictional practice; our map has no sales data.
36. Median of selected prizes
Level 2
The middle sorted prize limits the effect of a single huge value.
Worked example
Prizes 2,3,20 million have median 3 million.
Watch out
A selection median does not describe all players.
37. Identifiers and duplicate rows
Level 2
Excess rows equal all rows minus unique events.
Worked example
12 rows describe 9 events: 3 excess rows.
Watch out
Use date, game and identity, not amount alone.
38. Mean monthly count
Level 2
Add monthly counts and divide by month count.
Worked example
2,4,6 events: average 4.
Watch out
Missing month data is not a zero month.
39. Count after exclusion
Level 2
Subtract excluded records from the pre-filter count.
Worked example
Exclude 6 of 20 records: 14 remain.
Watch out
Do not subtract a record twice if it meets two exclusions.
40. Publication delay
Level 2
The gap between draw and publication dates describes reporting delay.
Worked example
Draw on day 4, publication on day 7: 3 days.
Watch out
Use an actual calendar when crossing month boundaries.
41. Region percentage
Level 2 · Regional shares
A share refers to a specified dataset and filter. Percentage is proportion times 100; disjoint parts sum to the whole.
Worked example
Teaching example. 4 of 50 records. Percentage? 4/50 × 100 = 8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
42. Count from percentage
Level 2 · Regional shares
A share refers to a specified dataset and filter. Percentage is proportion times 100; disjoint parts sum to the whole.
Worked example
Teaching example. 4% of 200 records. Count? 4/100 × 200 = 8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
43. Other regions
Level 2 · Regional shares
A share refers to a specified dataset and filter. Percentage is proportion times 100; disjoint parts sum to the whole.
Worked example
Teaching example. A has 4% of the dataset. Remaining percentage? 100 − 4 = 96
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
44. Two disjoint parts
Level 2 · Regional shares
A share refers to a specified dataset and filter. Percentage is proportion times 100; disjoint parts sum to the whole.
Worked example
Teaching example. A: 4%, B: 3%. Combined percentage? 4 + 3 = 7
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
45. Percentage-point change
Level 2 · Regional shares
A share refers to a specified dataset and filter. Percentage is proportion times 100; disjoint parts sum to the whole.
Worked example
Teaching example. Share from 4% to 7%. Change in pp? 7 − 4 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
46. Events per km²
Level 2 · Spatial density
Density is event count divided by area. It describes an area dataset, not the chance of an individual ticket.
Worked example
Teaching example. 40 events in 10 km². Density? 40/10 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
47. Recover the count
Level 2 · Spatial density
Density is event count divided by area. It describes an area dataset, not the chance of an individual ticket.
Worked example
Teaching example. Density 4/km², area 3 km². Count? 4 × 3 = 12
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
48. Recover the area
Level 2 · Spatial density
Density is event count divided by area. It describes an area dataset, not the chance of an individual ticket.
Worked example
Teaching example. 12 events at 4/km². Area km²? 12/4 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
49. Area conversion
Level 2 · Spatial density
Density is event count divided by area. It describes an area dataset, not the chance of an individual ticket.
Worked example
Teaching example. 4 km²: how many hectares? 4 × 100 = 400
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
50. Merged areas
Level 2 · Spatial density
Density is event count divided by area. It describes an area dataset, not the chance of an individual ticket.
Worked example
Teaching example. A: 20 events/10 km², B: 40/30 km². Combined density? (20+40)/(10+30) = 1.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
51. Per thousand tickets
Level 2 · Comparison denominators
Compare rates using matching denominators and periods. Population is not ticket sales.
Worked example
Teaching example. 4 hits per 2000 tickets. Per 1000? 4/2000 × 1000 = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
52. Per hundred thousand residents
Level 2 · Comparison denominators
Compare rates using matching denominators and periods. Population is not ticket sales.
Worked example
Teaching example. 4 events per 50000 residents. Per 100000? 4/50000 × 100000 = 8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
53. Rate ratio
Level 2 · Comparison denominators
Compare rates using matching denominators and periods. Population is not ticket sales.
Worked example
Teaching example. A: 4/1000, B: 2/1000. A/B? 4/2 = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
54. Rate difference
Level 2 · Comparison denominators
Compare rates using matching denominators and periods. Population is not ticket sales.
Worked example
Teaching example. A: 7/1000, B: 4/1000. Difference per 1000? 7 − 4 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
55. Pooled rate
Level 2 · Comparison denominators
Compare rates using matching denominators and periods. Population is not ticket sales.
Worked example
Teaching example. A: 2/1000, B: 8/4000. Pooled per 1000? (2+8)/(1000+4000) × 1000 = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
56. Absolute increase
Level 2 · Comparisons over time
Differences retain the count unit; percentage change divides the difference by the starting value. A zero baseline has no ordinary percentage change.
Worked example
Teaching example. 40 → 43 records. Increase? 43 − 40 = 3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
57. Percentage increase
Level 2 · Comparisons over time
Differences retain the count unit; percentage change divides the difference by the starting value. A zero baseline has no ordinary percentage change.
Worked example
Teaching example. Count 20 → 30. Percentage increase? (30−20)/20 × 100 = 50
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
58. Percentage decrease
Level 2 · Comparisons over time
Differences retain the count unit; percentage change divides the difference by the starting value. A zero baseline has no ordinary percentage change.
Worked example
Teaching example. Count 40 → 30. Magnitude of decrease in %? (40−30)/40 × 100 = 25
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
59. Base index
Level 2 · Comparisons over time
Differences retain the count unit; percentage change divides the difference by the starting value. A zero baseline has no ordinary percentage change.
Worked example
Teaching example. Base 20 = index 100. Index for 30? 30/20 × 100 = 150
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
60. Equal periods
Level 2 · Comparisons over time
Differences retain the count unit; percentage change divides the difference by the starting value. A zero baseline has no ordinary percentage change.
Worked example
Teaching example. 12 records/3 months and 20/5 months. Difference of monthly averages? 12/3 − 20/5 = 0
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
61. Largest-prize share
Level 3
Divide the largest amount by the selected total.
Worked example
Largest prize 6 million from total 10 million gives 60%.
Watch out
One dominant prize affects the mean.
62. Periods of different lengths
Level 3
An events-per-month rate divides count by period length.
Worked example
12 events over 3 months gives 4 per month.
Watch out
This describes the period, not the next month.
63. Intersection of filters
Level 3
Game AND year retains records meeting both conditions.
Worked example
8 game-A records, 3 from the year: result 3.
Watch out
AND and OR produce different sets.
64. Union of filters
Level 3
For OR, add counts and subtract the overlap.
Worked example
8 in A, 6 in B, 2 shared: 12 records.
Watch out
Identify shared records before adding.
65. Selection versus full registry
Level 3
Coverage requires a known complete event count.
Worked example
A fictional full registry has 40 events; selecting 10 covers 25%.
Watch out
For our selection without a full denominator, coverage is unknown.
66. Publication delay
Level 3 · Dates and observation windows
Separate event date from publication date. Specify inclusive boundaries and compare rates in matching time units.
Worked example
Teaching example. Event May 3, published May 7 of the same year. Elapsed days? 7 − 3 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
67. Inclusive interval
Level 3 · Dates and observation windows
Separate event date from publication date. Specify inclusive boundaries and compare rates in matching time units.
Worked example
Teaching example. Days 1–4 inclusive. How many days? 4 − 1 + 1 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
68. Daily average
Level 3 · Dates and observation windows
Separate event date from publication date. Specify inclusive boundaries and compare rates in matching time units.
Worked example
Teaching example. 28 events in 7 days. Daily average? 28/7 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
69. Sum of windows
Level 3 · Dates and observation windows
Separate event date from publication date. Specify inclusive boundaries and compare rates in matching time units.
Worked example
Teaching example. Two non-overlapping weeks: 4 and 3 events. Total? 4 + 3 = 7
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
70. Overlapping windows
Level 3 · Dates and observation windows
Separate event date from publication date. Specify inclusive boundaries and compare rates in matching time units.
Worked example
Teaching example. Windows contain 20 and 15 events, with 5 shared. Unique union? 20+15−5 = 30
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
71. Mappable count
Level 3 · Missing locations
Missing coordinates are not zero events. Report mappable counts and missingness separately.
Worked example
Teaching example. 100 records, 4 without location. Mappable? 100−4 = 96
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
72. Missing percentage
Level 3 · Missing locations
Missing coordinates are not zero events. Report mappable counts and missingness separately.
Worked example
Teaching example. 3 missing out of 50. Percentage? 3/50 × 100 = 6
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
73. Coordinate coverage
Level 3 · Missing locations
Missing coordinates are not zero events. Report mappable counts and missingness separately.
Worked example
Teaching example. 45 of 60 records have coordinates. Coverage %? 45/60 × 100 = 75
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
74. Completed locations
Level 3 · Missing locations
Missing coordinates are not zero events. Report mappable counts and missingness separately.
Worked example
Teaching example. There were 4 missing, 2 filled. Remaining? 4−2 = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
75. Coverage improvement
Level 3 · Missing locations
Missing coordinates are not zero events. Report mappable counts and missingness separately.
Worked example
Teaching example. Coverage 70% → 85%. Increase in pp? 85−70 = 15
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
76. Union of two sources
Level 3 · Combining sources
Join events by stable identifiers. Adding source counts overstates totals when the same events appear in multiple sources.
Worked example
Teaching example. A: 20 IDs, B: 12 IDs, 7 shared. Union? 20+12−7 = 25
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
77. Only in first
Level 3 · Combining sources
Join events by stable identifiers. Adding source counts overstates totals when the same events appear in multiple sources.
Worked example
Teaching example. A has 20 IDs, 7 also in B. Only A? 20−7 = 13
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
78. Redundant rows
Level 3 · Combining sources
Join events by stable identifiers. Adding source counts overstates totals when the same events appear in multiple sources.
Worked example
Teaching example. 60 rows, 45 unique IDs. Redundant rows? 60−45 = 15
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
79. Primary-source coverage
Level 3 · Combining sources
Join events by stable identifiers. Adding source counts overstates totals when the same events appear in multiple sources.
Worked example
Teaching example. 80 records, 60 with a primary source. Percentage? 60/80 × 100 = 75
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
80. Consistent field pairs
Level 3 · Combining sources
Join events by stable identifiers. Adding source counts overstates totals when the same events appear in multiple sources.
Worked example
Teaching example. 40 record pairs, 3 amount conflicts. Conflict-free? 40−3 = 37
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
81. Mean coordinate
Level 3 · Centres and weights
On a flat grid, a weighted coordinate mean is the sum of products divided by total weight. This is not automatically a geographic centre on a sphere.
Worked example
Teaching example. x: 2, 4, 6 km. Mean x? (2+4+6)/3 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
82. Weighted x
Level 3 · Centres and weights
On a flat grid, a weighted coordinate mean is the sum of products divided by total weight. This is not automatically a geographic centre on a sphere.
Worked example
Teaching example. x: 0, 10 km; weights 1, 3. Mean x? (0×1+10×3)/4 = 7.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
83. Weighted y
Level 3 · Centres and weights
On a flat grid, a weighted coordinate mean is the sum of products divided by total weight. This is not automatically a geographic centre on a sphere.
Worked example
Teaching example. y: 2, 8 km; weights 2, 1. Mean y? (2×2+8×1)/3 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
84. Translation effect
Level 3 · Centres and weights
On a flat grid, a weighted coordinate mean is the sum of products divided by total weight. This is not automatically a geographic centre on a sphere.
Worked example
Teaching example. All x values increase by 4 km. Increase of mean? 4 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
85. Zero weight
Level 3 · Centres and weights
On a flat grid, a weighted coordinate mean is the sum of products divided by total weight. This is not automatically a geographic centre on a sphere.
Worked example
Teaching example. x=2 weight 3, x=100 weight 0. Weighted mean? (2×3+100×0)/3 = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
86. Class width
Level 3 · Legend classes
Class boundaries must be unambiguous and non-overlapping. Changing classes changes map appearance, not data. The final interval may include its upper endpoint.
Worked example
Teaching example. Range 0–100 split into 5 equal classes. Width? (100−0)/5 = 20
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
87. Class number
Level 3 · Legend classes
Class boundaries must be unambiguous and non-overlapping. Changing classes changes map appearance, not data. The final interval may include its upper endpoint.
Worked example
Teaching example. Classes numbered from 1: [0,10), [10,20), [20,30]. Class for 10? 2 = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
88. Interval count
Level 3 · Legend classes
Class boundaries must be unambiguous and non-overlapping. Changing classes changes map appearance, not data. The final interval may include its upper endpoint.
Worked example
Teaching example. Data 0, 5, 10, 15, 20. Count in [10,20)? 2 = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
89. Upper endpoint
Level 3 · Legend classes
Class boundaries must be unambiguous and non-overlapping. Changing classes changes map appearance, not data. The final interval may include its upper endpoint.
Worked example
Teaching example. Final class [20,30]. Is 30 included? 1=yes, 0=no. 1 = 1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
90. Equal counts
Level 3 · Legend classes
Class boundaries must be unambiguous and non-overlapping. Changing classes changes map appearance, not data. The final interval may include its upper endpoint.
Worked example
Teaching example. 20 distinct values, 4 equal-count classes. Values per class? 20/4 = 5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
91. Fourfold count
Level 4 · Symbol area and radius
When circle area represents a count, radius grows with the square root of the count. Doubling radius multiplies area by four.
Worked example
Teaching example. Count increases fourfold, proportional area. Radius factor? √4 = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
92. Ninefold count
Level 4 · Symbol area and radius
When circle area represents a count, radius grows with the square root of the count. Doubling radius multiplies area by four.
Worked example
Teaching example. 10 events use radius 3 mm. Radius in mm for 90 events? 3 × √(90/10) = 9
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
93. Double radius
Level 4 · Symbol area and radius
When circle area represents a count, radius grows with the square root of the count. Doubling radius multiplies area by four.
Worked example
Teaching example. Radius is twice as large. Area factor? 2² = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
94. Half the radius
Level 4 · Symbol area and radius
When circle area represents a count, radius grows with the square root of the count. Doubling radius multiplies area by four.
Worked example
Teaching example. Radius halves. What fraction of area remains? (1/2)² = 0.25
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
95. Recovering the count
Level 4 · Symbol area and radius
When circle area represents a count, radius grows with the square root of the count. Doubling radius multiplies area by four.
Worked example
Teaching example. Radius 2 mm means 8 events. How many for radius 6 mm? 8 × (6/2)² = 72
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
96. Leading share
Level 4 · Concentration and aggregation
Concentration depends on area partitioning. Sum squared shares using proportions 0–1; do not confuse this with future winning probability.
Worked example
Teaching example. Counts 8, 6, 4, 2. Leading share as a fraction? 8/20 = 0.4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
97. Top-two share
Level 4 · Concentration and aggregation
Concentration depends on area partitioning. Sum squared shares using proportions 0–1; do not confuse this with future winning probability.
Worked example
Teaching example. Counts 8, 6, 4, 2. Top-two share as a fraction? (8+6)/20 = 0.7
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
98. Equal partition
Level 4 · Concentration and aggregation
Concentration depends on area partitioning. Sum squared shares using proportions 0–1; do not confuse this with future winning probability.
Worked example
Teaching example. Four regions each have 1/4. Sum of squared shares? 4 × (1/4)² = 0.25
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
99. Unequal partition
Level 4 · Concentration and aggregation
Concentration depends on area partitioning. Sum squared shares using proportions 0–1; do not confuse this with future winning probability.
Worked example
Teaching example. Two regions: 0.75 and 0.25. Sum of squared shares? 0.75²+0.25² = 0.625
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
100. After merging boundaries
Level 4 · Concentration and aggregation
Concentration depends on area partitioning. Sum squared shares using proportions 0–1; do not confuse this with future winning probability.
Worked example
Teaching example. Four equal regions merged in pairs. Sum of squares of the two new shares? 0.5²+0.5² = 0.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
101. Minimum count
Level 4 · Bounds with missing data
Rather than assigning missing locations randomly, calculate extreme possible shares. These are logical bounds, not confidence intervals.
Worked example
Teaching example. 12 events known in A; 8 with no region. Minimum in A? 12 = 12
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
102. Maximum count
Level 4 · Bounds with missing data
Rather than assigning missing locations randomly, calculate extreme possible shares. These are logical bounds, not confidence intervals.
Worked example
Teaching example. 12 known in A; 8 without region. Maximum in A? 12+8 = 20
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
103. Minimum share
Level 4 · Bounds with missing data
Rather than assigning missing locations randomly, calculate extreme possible shares. These are logical bounds, not confidence intervals.
Worked example
Teaching example. Total 50, A known 12, unknown region 8. Minimum A share %? 12/50 × 100 = 24
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
104. Maximum share
Level 4 · Bounds with missing data
Rather than assigning missing locations randomly, calculate extreme possible shares. These are logical bounds, not confidence intervals.
Worked example
Teaching example. Total 50, A known 12, unknown region 8. Maximum A share %? 20/50 × 100 = 40
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
105. Width of bounds
Level 4 · Bounds with missing data
Rather than assigning missing locations randomly, calculate extreme possible shares. These are logical bounds, not confidence intervals.
Worked example
Teaching example. Share lies within 24–40%. Width in pp? 40−24 = 16
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
106. First-group rate
Level 4 · Groups and pooled comparisons
Pooling rates requires denominator weights. Aggregate data do not assign risk to each person; group composition can change comparisons.
Worked example
Teaching example. Group A: 8/10. Percentage? 8/10 × 100 = 80
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
107. Second-group rate
Level 4 · Groups and pooled comparisons
Pooling rates requires denominator weights. Aggregate data do not assign risk to each person; group composition can change comparisons.
Worked example
Teaching example. Group B: 18/90. Percentage? 18/90 × 100 = 20
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
108. Overall rate
Level 4 · Groups and pooled comparisons
Pooling rates requires denominator weights. Aggregate data do not assign risk to each person; group composition can change comparisons.
Worked example
Teaching example. A: 8/10, B: 18/90. Overall percentage? (8+18)/(10+90) × 100 = 26
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
109. Unweighted average
Level 4 · Groups and pooled comparisons
Pooling rates requires denominator weights. Aggregate data do not assign risk to each person; group composition can change comparisons.
Worked example
Teaching example. Arithmetic mean of 80% and 20%? Give percentage. (80+20)/2 = 50
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
110. Incorrect-weighting difference
Level 4 · Groups and pooled comparisons
Pooling rates requires denominator weights. Aggregate data do not assign risk to each person; group composition can change comparisons.
Worked example
Teaching example. Unweighted mean 50%, pooled rate 26%. Difference in pp? 50−26 = 24
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
111. One-axis error
Level 4 · Precision and rounding
Display precision does not guarantee location accuracy. Rounding to step d gives error at most d/2 on one axis when rounding to the nearest value.
Worked example
Teaching example. Rounded to 100 m. Maximum axis error in m? 100/2 = 50
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
112. Two-axis error
Level 4 · Precision and rounding
Display precision does not guarantee location accuracy. Rounding to step d gives error at most d/2 on one axis when rounding to the nearest value.
Worked example
Teaching example. Axis errors 3 m and 4 m. Resulting displacement in m? √(3²+4²) = 5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
113. Interval span
Level 4 · Precision and rounding
Display precision does not guarantee location accuracy. Rounding to step d gives error at most d/2 on one axis when rounding to the nearest value.
Worked example
Teaching example. Distance 8–12 km. Span in km? 12−8 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
114. Interval midpoint
Level 4 · Precision and rounding
Display precision does not guarantee location accuracy. Rounding to step d gives error at most d/2 on one axis when rounding to the nearest value.
Worked example
Teaching example. Distance 8–12 km. Midpoint km? (8+12)/2 = 10
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
115. Relative error
Level 4 · Precision and rounding
Display precision does not guarantee location accuracy. Rounding to step d gives error at most d/2 on one axis when rounding to the nearest value.
Worked example
Teaching example. Absolute error 2 km relative to 40 km. Percentage? 2/40 × 100 = 5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
116. Clean dataset
Level 4 · Project: map audit
Before interpreting a map, check unique events, sources, units, locations and period. Record exclusions; missing data are not proof of no events.
Worked example
Teaching example. 100 rows, 10 redundant duplicate rows. Unique events? 100−10 = 90
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
117. Presentation dataset
Level 4 · Project: map audit
Before interpreting a map, check unique events, sources, units, locations and period. Record exclusions; missing data are not proof of no events.
Worked example
Teaching example. 90 unique events, 15 without location. Mappable? 90−15 = 75
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
118. Project filter
Level 4 · Project: map audit
Before interpreting a map, check unique events, sources, units, locations and period. Record exclusions; missing data are not proof of no events.
Worked example
Teaching example. 75 located, 20 outside the period. Visible count? 75−20 = 55
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
119. Source-completeness check
Level 4 · Project: map audit
Before interpreting a map, check unique events, sources, units, locations and period. Record exclusions; missing data are not proof of no events.
Worked example
Teaching example. 55 visible, 44 with checked sources. Percentage? 44/55 × 100 = 80
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
120. Legend-total check
Level 4 · Project: map audit
Before interpreting a map, check unique events, sources, units, locations and period. Record exclusions; missing data are not proof of no events.
Worked example
Teaching example. 55 visible events. Legend classes: 10, 15, 20. How many unaccounted for? 55−(10+15+20) = 10
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
Test: choose your scope
Choose a lesson range and test length. Topics are drawn without repetition, up to the number available in that range. Answers appear after the test; there is no time limit.
Further reading
Original exercises by Pewniaki. The mental-maths identities are explained algebraically; no claim of historical origin or guaranteed speed advantage is made.