Connecting the Dots... · Lesson 11 of 13
Data Detective
“Investigate height patterns while checking samples, predictions, and the limits of averages.”
• Read a multi-year age-group height table. • Compare historical and adjacent-age differences. • Recognise limits of school samples and group averages. • Distinguish data-supported conclusions from estimates and predictions. • Interpret nonzero graph baselines and choose measures suited to a decision.
Data Detective
A detective looks for patterns, checks evidence, and asks what remains unknown. A data detective does the same with observations and graphs. This final teaching investigation connects school-height comparisons with age patterns and historical changes, while keeping group averages separate from individual claims.
Telling Tall Tales
The source compares boys and girls in Grades 6, 7, and 8 at two schools. Within each school, the plots overlap, and their centres vary by grade. School B’s recorded mean heights are greater in these comparisons. That observation invites a question about why the schools differ; it is not an explanation of the difference.
| Group | School A mean (cm) | School B mean (cm) |
|---|---|---|
| Grade 6 boys | 134.8 | 149.84 |
| Grade 6 girls | 137.78 | 150.2 |
| Grade 7 boys | 141.8 | 156.14 |
| Grade 7 girls | 141.83 | 155.41 |
| Grade 8 boys | 149.35 | 156.14 |
| Grade 8 girls | 147.81 | 156.83 |
In School A, Grade 6 girls have a greater mean than boys, while Grade 8 boys have a greater mean than girls. In School B, the Grade 8 girls have a slightly greater mean than boys. These changing orders show why one classroom cannot support a rule about all children in a country. Individual heights also overlap even when the means differ.
Problem
Compare Grade 6 boys in the two schools.
- 1.School A’s supplied mean is 134.8 cm; School B’s is 149.84 cm.
- 2.Difference = 149.84 − 134.8 = 15.04 cm.
- 3.This is a comparison of two measured groups. It does not identify their locations, prove a cause, or show that every boy at B is taller than every boy at A.
A collection from one or two schools is a sample of a much wider population. Its composition may differ from other schools. We should ask which people were measured and how they were selected before extending the finding. Knowing subgroup means without their observation counts also does not determine an overall mean.
Reading heights across ages and years
The next table supplies average heights in India for ages 5–19 in 1989, 1999, 2009, and 2019. Each year has separate boys’ and girls’ columns. Read the age, year, group, and centimetre unit together before choosing a value.
| Age | 1989 boys | 1989 girls | 1999 boys | 1999 girls | 2009 boys | 2009 girls | 2019 boys | 2019 girls |
|---|---|---|---|---|---|---|---|---|
| 5 | 101.3 | 100 | 102.4 | 101.7 | 105.1 | 104 | 107.1 | 107.2 |
| 6 | 107.5 | 106 | 108.7 | 107.5 | 111 | 109.7 | 113.1 | 112.9 |
| 7 | 113 | 111.4 | 114.2 | 112.6 | 116.2 | 114.8 | 118.6 | 118 |
| 8 | 118.1 | 116.5 | 119.2 | 117.5 | 120.9 | 119.6 | 123.5 | 122.7 |
| 9 | 122.9 | 121.7 | 123.9 | 122.4 | 125.2 | 124.5 | 128.1 | 127.6 |
| 10 | 127.5 | 127.3 | 128.3 | 127.8 | 129.4 | 129.9 | 132.6 | 132.8 |
| 11 | 132.2 | 133.4 | 132.8 | 133.6 | 133.7 | 135.7 | 137 | 138.6 |
| 12 | 137.7 | 139 | 138 | 139.1 | 138.9 | 141.1 | 142.2 | 143.8 |
| 13 | 144.2 | 143.2 | 144.3 | 143.1 | 145.2 | 145.1 | 148.4 | 147.7 |
| 14 | 150.6 | 146.2 | 150.5 | 146.1 | 151.5 | 148 | 154.4 | 150.4 |
| 15 | 155.4 | 148.5 | 155.2 | 148.4 | 156.3 | 150.1 | 159 | 152.4 |
| 16 | 158.9 | 150.1 | 158.7 | 150.1 | 159.9 | 151.6 | 162.3 | 153.8 |
| 17 | 161.3 | 151.2 | 161.4 | 151.3 | 162.6 | 152.6 | 164.6 | 154.7 |
| 18 | 162.9 | 151.8 | 163.2 | 152.1 | 164.3 | 153 | 166 | 155.2 |
| 19 | 163.5 | 151.9 | 164.2 | 152.4 | 165.1 | 153 | 166.5 | 155.2 |
Move across a row to compare the same age through time. Move down one column to compare ages in one survey year. Compare the boys’ and girls’ columns for the same year and age to study their group averages. These are average heights, not the height of every child.
Problem
How did the supplied mean height of nineteen-year-old boys change from 1989 to 2019?
- 1.Read age 19 in the boys’ columns: 163.5 cm in 1989 and 166.5 cm in 2019.
- 2.Subtract: 166.5 − 163.5 = 3 cm.
- 3.The supplied average increased by 3 cm across those years. The table does not mean the same individuals were measured thirty years apart.
Problem
Between which successive ages is the largest increase for boys in the 2019 column?
- 1.Subtract each age’s mean from the next age’s mean. Between 12 and 13, the difference is 148.4 − 142.2 = 6.2 cm.
- 2.The other successive differences are no larger; the 5-to-6 difference is 6.0 cm and the 13-to-14 difference is also 6.0 cm.
- 3.Thus 12–13 has the largest difference in these supplied age-group means. This is not a tracked individual child’s growth rate.
For girls in 2019, the largest adjacent-age difference is from 5 to 6: 112.9 − 107.2 = 5.7 cm. Girls’ mean heights exceed boys’ at several ages, while boys’ mean is higher at older ages. A blanket claim that boys have the greater mean throughout ages 5–19 is contradicted by the table.
Checking statements against evidence
A claim can be supported, contradicted, or not answerable from the supplied information. Do not force every unsupported claim into “false in the real world”. Sometimes the data simply does not measure what would be needed to decide it.
| Claim | Assessment from this table |
|---|---|
| Both groups’ averages at every listed age increased from 1989 to 2019. | Supported by the two endpoint columns. |
| The 1989 mean for 13-year-old girls exceeds the 2009 mean for 14-year-old girls. | Contradicted: 143.2 < 148. |
| The 2019 mean for 15-year-old boys exceeds the 1989 mean for 16-year-old boys. | Supported: 159 > 158.9. |
| Every thirteen-year-old girl is taller than every eleven-year-old girl. | Not established by averages. Individual distributions are not supplied. |
| Boys have a greater mean than girls at every listed age. | Contradicted by ages such as 5, 10, 11, and 12 in 2019. |
| Boys continue growing beyond nineteen. | Not answerable: ages beyond nineteen are absent. |
Averages at successive ages may describe different groups of children. Do not turn their differences into a guaranteed growth path for an individual or use group averages to judge one child.
Estimates, predictions, and international comparisons
The source asks readers to estimate missing ages and future years. Those are opportunities to reason from a pattern, not invitations to invent measurements. State the assumptions, and distinguish a prediction from an observation.
Problem
If a newborn mean is assumed to be 50 cm and a five-year-old mean is 107.1 cm, what simple estimates could a constant-increase model give for ages 1–4?
- 1.The five-year difference is 107.1 − 50 = 57.1 cm.
- 2.If we assume equal yearly increases, each increase is 57.1 ÷ 5 = 11.42 cm.
- 3.That model gives 61.42, 72.84, 84.26, and 95.68 cm at ages 1–4.
- 4.These are illustrative model estimates, not supplied measurements or a claim that early growth is actually constant. Other justified assumptions give different estimates.
Problem
Use the latest decade’s change to suggest 2029 mean heights for nineteen-year-olds.
- 1.Boys increased from 165.1 to 166.5 between 2009 and 2019: a change of 1.4 cm. Girls increased from 153 to 155.2: 2.2 cm.
- 2.If those changes were repeated for another decade, the estimates would be 167.9 cm for boys and 157.4 cm for girls.
- 3.Label these as predictions under a repeated-change assumption, not actual 2029 data.
The international graph compares nineteen-year-old boys and girls in 1989 and 2019 across Timor-Leste, Yemen, Bangladesh, Liberia, Indonesia, India, Bhutan, Pakistan, Maldives, Kenya, Saudi Arabia, Japan, Sudan, Taiwan, Italy, Egypt, South Korea, China, Israel, Turkey, Libya, New Zealand, the UK, Austria, Norway, Ukraine, Iceland, and the Netherlands. Crosses identify boys and triangles identify girls; colours distinguish years. Read all four series from the legend.
Its vertical axis starts at 145 cm and extends to 185 cm. That zoom helps readers distinguish nearby heights. The height of a plotted point is a numerical position, not a bar length starting at zero. A point appearing twice as far above 145 does not represent twice the person’s height. Across the displayed countries, both country differences and historical changes can be observed, but the graph does not explain their causes.
Problem
On an axis starting at 145 cm, compare points at 155 and 175 cm.
- 1.They stand 10 and 30 cm above the plotted baseline. The second visible offset is three times the first.
- 2.Their actual height ratio is 175 ÷ 155 ≈ 1.13, not three.
- 3.Use axis values for ratios. A nonzero starting value changes visible offsets, not the underlying measurements.
A Mean Decision!
An average is useful only when it fits the decision. If a doorway must allow every member of a family through comfortably, the family’s mean height is not enough. Some members will be taller than the mean, and a doorway at that height would not serve them.
For this decision, the tallest relevant person and suitable clearance matter. For a description of typical family height, mean or median can still be useful. The lesson is to choose a measure according to purpose rather than treating one measure as universally best.
Quiz
What do the source height-table values represent?
In 2019, between ages 12 and 13, boys’ supplied means differ by:
Can one school’s mean determine another school’s mean?
What can the table say about growth beyond age 19?
A 2029 estimate based on a repeated-decade change is:
For a doorway intended for all family members, which information is essential?
Practice Problems
- Compare 2019 mean heights of boys and girls aged 11.
- Is it true that every decade-to-decade height change is an increase at every listed age?
- Two Grade 7 sections each have fifteen boys and fifteen girls. One mean is 154.2 cm. What must be the other section’s mean?
- Use the Indian table to discuss the likely ages of students with heights around 101–125 cm.
- Explain why a height graph starting at 145 cm must not be used to infer ratios from visible offsets.
- Choose an age and predict 2029 from its 2009–2019 change. State the assumption.
- What evidence would you need to explain the difference between two schools’ height distributions?
Boys: 137 cm; girls: 138.6 cm. Girls’ mean is higher by 1.6 cm. This does not mean every girl exceeds every boy.
Key Takeaways
• Read the age, year, group, and unit together. • Sample comparisons cannot automatically be generalised to all people. • Different age-group averages are not an individual growth record. • Separate supported, contradicted, and unanswerable claims. • Label model estimates and predictions with their assumptions. • A representative value must fit the decision being made.