Let's break down and address each part of Task 5 according to the provided data regarding measures of central tendency and range.
a) Compare your class data to the sampling of the same size (4th column) in terms of measures of central tendency and range.
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Mean:
- My Class Data: 55.45
- Sample 4: 45.05
- Comparison: My class has a higher mean (55.45) compared to the sample of all Year 8 students (45.05), suggesting that my class performed better on average.
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Mode:
- My Class Data: 50, 49, 58
- Sample 4: 26
- Comparison: The modes of my class data indicate multiple common scores (50, 49, 58), while the sample has a single mode (26), suggesting a wider spread of performance in my class.
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Median:
- My Class Data: 54
- Sample 4: 41
- Comparison: The median of my class (54) is higher than that of the overall Year 8 sample (41), indicating that the data point in the middle of my class’s distribution is greater.
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Range:
- My Class Data: 53
- Sample 4: 89
- Comparison: My class has a smaller range (53) compared to the Year 8 sample (89), meaning that scores in my class are more clustered together relative to the wider spread of scores in the Year 8 population.
b) Compare the variation between the measures of central tendency and range when the sample size was changed.
When the sample size increases from my class (10 students) to larger samples (20 and 100 students), and eventually to the entire Year 8 cohort:
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Mean: The mean varied significantly across different sample sizes, with my class yielding the highest mean at 55.45, suggesting a trend that smaller, more homogeneous groups can have higher means due to limited variation.
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Mode: The modes show considerable variation, reflecting differences in score distribution. Smaller groups can have multiple modes while larger groups may consolidate towards fewer dominant scores.
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Median: The median values also reflect the increase in diversity and distribution of scores; increased sample size likely provides a more accurate representative median for the broader population.
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Range: The range considerably increased with sample size, indicating that as more students are included, the performance spread broadens, showing variability in skill levels.
c) In order to determine if your class is typical, select a sample that best represents what typical is for Year 8 students. Justify your choice by referring to the selected sample size.
Selected Sample: Sample 2 (20 students)
Justification: Sample 2 represents a middle-ground option in terms of size that is less likely to be skewed by outlier performances. Comparing my class of 10 students to a slightly larger sample (20 students) allows for a more balanced reflection of the broader Year 8 population while still being manageable for analysis. This sample size provides enough data to capture variability but is still small enough to maintain relevance to my class’s performance.
d) Compare your class to the sample size you have selected to determine whether our class is typical or not.
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Mean Comparison: My class mean (55.45) is notably higher than Sample 2's mean (46). This suggests that my class is performing above what may be typical for Year 8 students.
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Mode Comparison: My class's modes show a more concentrated performance (values close to each other) than Sample 2's mode (32, 29), indicating my class may have a better skill level or cohesive performance.
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Median Comparison: My class median (54) significantly exceeds the median of Sample 2 (36), reinforcing the notion that my class is performing above the average of the sampled Year 8 population.
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Range Comparison: The smaller range in my class (53) compared to Sample 2's range (64) suggests greater uniformity in performance within my class compared to a wider variation in Sample 2, indicating that the broader Year 8 population has more extreme scores.
e) Reflect on the reasonableness of your findings. Was the comparison carried out reasonable in determining whether our class typical typers? Why/why not?
The comparisons made were reasonable as they utilized various measures of central tendency to assess performance across different sample sizes. Utilizing means, modes, and medians provides a robust framework for evaluation, while ranges help illustrate the distribution of scores.
However, the results may be influenced by sample selection and potential outliers. My class may possess unique characteristics (e.g., coaching, motivation, academic programs) that set it apart from a more representative cohort of Year 8 students. Additional context or qualitative data might be necessary to fully determine whether my class is truly typical. It’s crucial to consider external influences that could have affected performance, ensuring that the analysis doesn't just rely on raw scores but also reflects broader educational contexts.