The World Health Organization (WHO) has produced a comprehensive set of growth standards for children. These standards are based on...

OWNING THE DATASET

Understanding Source A: Table - WHO Height-for-Age Standards for Boys Ages 2-5

Summary: The WHO table shows ideal height ranges for boys ages 2-5, with percentile values tracking how height varies in a healthy population at 3-month intervals.

Understanding Source B: Graph - WHO Weight-for-Height Standards for Boys Ages 2-5

Chart Analysis:

• The graph shows weight (in kilograms) plotted against height (\(\mathrm{80-120\,cm}\)) for boys ages 2-5
• Five curves represent the same percentiles as Source A: 3rd, 15th, 50th, 85th, and 97th
• Key patterns observed: All curves slope upward, showing weight increases with height
• Weight ranges from approximately \(\mathrm{9-13\,kg}\) at \(\mathrm{80\,cm}\) to \(\mathrm{18-26\,kg}\) at \(\mathrm{120\,cm}\)
• Inference: The spread between lightest and heaviest boys increases with height (roughly \(\mathrm{4\,kg}\) spread at \(\mathrm{80\,cm}\), expanding to roughly \(\mathrm{8\,kg}\) spread at \(\mathrm{120\,cm}\))
• Linkage to Source A: The height range \(\mathrm{80-120\,cm}\) directly corresponds to the typical heights shown in Source A's table for ages 2-5
• Linkage to Source A: Both sources show the same pattern of increasing variability - just as height differences widen with age in Source A, weight differences widen with height in Source B

Summary: Source B complements Source A by showing appropriate weights for each height, allowing complete growth assessment - Source A tells if a boy is the right height for his age, while Source B tells if he's the right weight for his height.

Overall Summary

• The WHO growth standards provide a comprehensive framework for assessing boys' development ages 2-5
• Together, these sources enable complete growth monitoring: Source A determines if a child's height is appropriate for their age, while Source B determines if their weight is appropriate for their height
• Both sources use identical percentile systems (3rd, 15th, 50th, 85th, 97th)
• Both sources show that individual differences in growth become more pronounced as children develop - whether measuring height-for-age or weight-for-height

Question Analysis

This analysis examines three statements about boys' height and weight relationships using two sources: height-for-age standards showing percentile distributions for boys ages 2-5, and weight-for-height standards displaying weight distribution curves for heights from 80-120 cm. Each statement requires evaluation against specific percentile thresholds and age-related growth patterns.

Connecting to Our Analysis

The evaluation utilizes percentile data to determine probabilities and comparative standings. Source A provides age-specific height percentiles at 3-month intervals, while Source B shows weight percentiles for different height measurements. The analysis requires cross-referencing between age-height relationships and height-weight correlations to assess each statement's validity.

Extracting Relevant Findings

Source A - Height-for-age standards: Provides percentile distribution of heights for boys ages 2-5 at 3-month intervals. Key insight: Heights increase with age, and we can determine what percentage of boys at each age are above or below specific height thresholds.

Source B - Weight-for-height standards: Shows weight distribution curves for different height values from 80-120 cm. The graph displays \(\mathrm{3^{rd}}\), \(\mathrm{15^{th}}\), \(\mathrm{50^{th}}\), \(\mathrm{85^{th}}\), and \(\mathrm{97^{th}}\) percentile lines for weight at each height.

Individual Statement/Option Evaluations

Statement 1 Evaluation

Statement: "If his age is greater than 3 years 3 months, the probability that his height is at least 98.0 cm is greater than 50%."

• Data Source: Source A height-for-age standards
• Key Reference Point: At age \(\mathrm{3:3}\), the \(\mathrm{50^{th}}\) percentile height = \(\mathrm{98.0\,cm}\)
• Critical Insight: This means exactly 50% of boys at \(\mathrm{3:3}\) are \(\mathrm{98.0\,cm}\) or taller
• Evidence for ages greater than 3:3:
  • Age \(\mathrm{3:6}\): \(\mathrm{50^{th}}\) percentile = \(\mathrm{99.9\,cm}\)
  • Age \(\mathrm{3:9}\): \(\mathrm{50^{th}}\) percentile = \(\mathrm{101.6\,cm}\)
  • Age \(\mathrm{4:0}\): \(\mathrm{50^{th}}\) percentile = \(\mathrm{103.3\,cm}\)
  • All subsequent ages show \(\mathrm{50^{th}}\) percentile heights above \(\mathrm{98.0\,cm}\)
• Reasoning: Since the median height for all ages greater than \(\mathrm{3:3}\) exceeds \(\mathrm{98.0\,cm}\), more than 50% of boys in these age groups are at least \(\mathrm{98.0\,cm}\) tall
• Conclusion: YES

Statement 2 Evaluation

Statement: "If he is at least 105 cm tall, the probability that his weight is 14.0 kg is no greater than 3%"

• Data Source: Source B weight-for-height standards
• Height Analysis: At height \(\mathrm{105\,cm}\)
• Critical Data Point: The \(\mathrm{3^{rd}}\) percentile line shows approximately \(\mathrm{14.5\,kg}\)
• Interpretation: Only 3% of boys at \(\mathrm{105\,cm}\) height weigh \(\mathrm{14.5\,kg}\) or less
• Key Insight: Since \(\mathrm{14.0\,kg}\) is below the \(\mathrm{3^{rd}}\) percentile weight, fewer than 3% of boys at this height would weigh \(\mathrm{14.0\,kg}\) or less
• Reasoning: The probability of weighing exactly \(\mathrm{14.0\,kg}\) at this height is indeed no greater than 3%
• Conclusion: YES

Statement 3 Evaluation

Statement: "If he is 114 cm tall, he is taller than at least 85% of boys his age."

• Data Source: Source A height-for-age standards
• Approach: Examine which ages could have boys at \(\mathrm{114\,cm}\) height
• Age-specific Analysis:
  • Age \(\mathrm{4:6}\): \(\mathrm{85^{th}}\) percentile = \(\mathrm{111.2\,cm}\), \(\mathrm{97^{th}}\) percentile = \(\mathrm{115.0\,cm}\) (\(\mathrm{114\,cm}\) is between these)
  • Age \(\mathrm{4:9}\): \(\mathrm{85^{th}}\) percentile = \(\mathrm{113.0\,cm}\), \(\mathrm{97^{th}}\) percentile = \(\mathrm{116.8\,cm}\) (\(\mathrm{114\,cm}\) is just above \(\mathrm{85^{th}}\))
  • Age \(\mathrm{5:0}\): \(\mathrm{85^{th}}\) percentile = \(\mathrm{114.8\,cm}\) (\(\mathrm{114\,cm}\) is BELOW the \(\mathrm{85^{th}}\) percentile)
• Critical Finding: For a boy aged \(\mathrm{5:0}\) who is \(\mathrm{114\,cm}\) tall, he would be shorter than the \(\mathrm{85^{th}}\) percentile
• Counterexample: This boy is NOT taller than 85% of boys his age
• Reasoning: Since the statement must be true for ANY boy who is \(\mathrm{114\,cm}\) tall, and we found a counterexample, the statement is false
• Conclusion: NO

Systematic Checking

Each evaluation was verified against the appropriate data source with careful attention to percentile interpretations and age-specific growth patterns. The analysis confirms that statements 1 and 2 are supported by the percentile data, while statement 3 fails due to age-dependent height variations that create scenarios where the claim does not hold universally.

Final Answer

• Statement 1: Yes
• Statement 2: Yes
• Statement 3: No