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Mathematics Applications & Interpretation (AI) IA Exemplar: Poisson… | RevisionDojo
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IB Mathematics Applications & Interpretation (AI) SL Internal Assessment Example
How can the Poisson distribution be applied to create a model and predict the probability of a successful number of penalty kicks in football, considering factors such as home or away games, time of the match, and the team’s winning or losing status?SL
4
Official IB Result
9/20
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Criteria A: Presentation
3/4
0
2
4
Criteria Strands
A.1Coherence and logical development
Good
A.2Organization and structure
Good
A.3Conciseness and relevance
Moderate
Criteria Feedback
Clear and logical overall structure with recognizable introduction, body and conclusion.
Effective use of headings, numbered figures and tables to guide the reader.
Coherent development of the argument through sections (aim → data → mathematics → model → conclusion).
Occasional rambling passages and digressions reduce conciseness.
Repetition of the aim and verbose narrative in places.
Inconsistent formatting of raw tables (e.g. Figure 12) affects clarity.
1.1·Strength
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The research question is clearly stated and sets a focused investigative framework, guiding the exploration of Poisson modelling in football penalties.
1.2·Suggestion
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The narrative in the introduction is verbose and sometimes repetitive. Consider tightening sentences and avoiding restating the aim to improve conciseness and relevance.
1.3·Strength
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The aim section succinctly outlines objectives for applying Poisson distribution and building a model, providing clear direction for the investigation.
1.4·Suggestion
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Figure 6’s working column is cluttered with multiple lines. Separate the working out from the final expected values to improve clarity.
1.5·Weakness
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The raw simulation results table (Figure 12) includes atypical symbols and inconsistent formatting. Simplify entries to standard numerals for clarity.
Criteria B: Mathematical Communication
1/4
0
2
4
Criteria Strands
B.1Mathematical language and notation
Moderate
B.2Multiple representations
Moderate
B.3Clarity and consistency
Moderate
Criteria Feedback
Correct presentation of the Poisson formula using λ and factorial notation.
Basic spreadsheet and table representations support the argument at key points.
Inconsistent use of mathematical notation (mixing percentages and probabilities).
Limited variety of representations – missing graphs or more sophisticated visuals.
Frequent typographical errors and deeply nested IF formulas hinder clarity.
2.1·Suggestion
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The calculation of λ is correct but percentages are mixed with probabilities. Standardize notation: report λ=94/649≈0.145 (14.5%).
2.2·Strength
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The Poisson formula is presented accurately and formatted clearly, supporting correct mathematical communication.
2.3·Weakness
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The empty probability table in Figure 4 lacks content. Complete the table with computed probabilities to make the representation effective.
2.4·Suggestion
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Figure 5’s table of λ values per condition mixes event rates and proportions without clear units. Clarify whether these λ are per‐game rates or proportions of success.
2.5·Suggestion
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The Excel IF formula for simulating penalty counts is functional but deeply nested and reduces readability. Consider using a lookup table or SWITCH for clarity.
2.6·Suggestion
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The long nested IF chain in your code snippet compromises mathematical communication. Suggest presenting a structured decision table instead.
Criteria C: Personal Engagement
2/3
0
2
3
Criteria Strands
C.1Independent thinking
Good
C.2Personal approach
Good
C.3Creativity and initiative
Good
Criteria Feedback
Clear personal motivation from a genuine interest in football.
Decision to collect bespoke data and adapt λ values shows independent thinking.
Designing and running a Monte-Carlo style simulation demonstrates initiative.
The personal narrative, while present, could be more tightly woven into the mathematical discussion.
Some choices (e.g. trial number, λ adjustments) lack commentary on alternative approaches.
3.1·Strength
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The introduction demonstrates outstanding personal engagement by linking the student’s long‐term interest in football to the mathematical investigation.
Criteria D: Reflection
1/3
0
2
3
Criteria Strands
D.1Depth of reflection
Good
D.2Critical analysis
Good
D.3Connection to understanding
Good
Criteria Feedback
Reflection on sampling bias and model limitations shows awareness.
Comments link the outcomes to understanding of Poisson assumptions.
Depth of reflection remains surface-level without thorough quantification of biases.
Critical analysis lacks statistical rigour and is not sustained throughout.
Connections to broader implications and validation are suggested but unfulfilled.
4.1·Suggestion
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The simulation summary lacks comparison with real‐world penalty frequencies. Include validation by comparing simulated averages with actual season data.
4.2·Suggestion
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The conclusion recognizes achievement of aims but lacks critical analysis of result deviations. Reflect on discrepancies between expected and simulated values.
4.3·Strength
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The reflection on convenience sampling and player selection bias is meaningful and shows awareness of methodological limitations.
4.4·Suggestion
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While various limitations are listed, the evaluation lacks quantification of their impact. Consider estimating how each bias could quantitatively alter your results.
Criteria E: Use of Mathematics
2/6
0
3
6
Criteria Strands
E.1Relevance and level
Moderate
E.2Accuracy and correctness
Poor
E.3Knowledge and understanding
Moderate
Criteria Feedback
Appropriate application of Poisson distribution and expected value calculations within SL syllabus.
Correct discussion of independence and discrete-event assumptions.
Basic Monte-Carlo simulation logic is logically applied.
Rounding inconsistencies and percentage/probability confusions introduce substantive errors.
Some computations (e.g. λ adjustments, impact calculations) lack full justification.
Missing summary statistics to highlight key trends from raw data.
5.1·Suggestion
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The raw data table is extensive but lacks summary statistics. Add a concise summary (e.g., totals and proportions) to highlight key trends and improve relevance.
5.2·Strength
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The discussion of Poisson assumptions effectively conveys concepts of independence and discrete events, demonstrating good understanding of underlying theory.
5.3·Weakness
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The interpretation “0.145% chance” miscues percentage and probability. Clarify that λ=0.145 corresponds to a 14.5% probability, not 0.145%.
5.4·Suggestion
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The formula for expected goals is introduced correctly, but lacks connection to the Poisson model derivation. Include a rationale linking EV=N·λ·p to Poisson expectations.
5.5·Weakness
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The computation of average games per season divides total games by three players and seasons without clear justification. Explain why this approach approximates N.
5.6·Weakness
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Impact calculations in Figure 9 treat percentage changes additively. In probabilistic models, apply multiplicative adjustments to preserve probability bounds.
