IB Mathematics AA Concepts of population and sample Study Notes
IB Mathematics AA Concepts of population and sample Study Notes
IB Mathematics AA Concepts of population and sample Notes Offer a clear explanation of Concepts of population and sample, including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Concepts of population and sample.
Concepts of Population & Sample
Introduction
The study of populations and samples is fundamental in statistics. This topic covers key concepts such as random sampling, data types, reliability, and the identification and interpretation of outliers. Understanding these ideas is essential for accurate data analysis and interpretation.
Key Concepts
Population and Sample
- Population: The complete set of items or individuals being studied in a statistical analysis.
- Sample: A smaller subset of the population selected for analysis.
- Random Sample: A sample chosen such that every member of the population has an equal chance of selection, ensuring representativeness.
Data Types
- Discrete Data: Consists of countable values (e.g., number of cars in a parking lot).
- Continuous Data: Can take any value within a range (e.g., temperature, height).
Reliability and Bias
- Reliability of Data Sources: The accuracy and trustworthiness of data depend on credible collection methods and sources.
- Bias in Sampling: Occurs when the sampling method introduces a systematic error, leading to an unrepresentative sample.
- Dealing with Missing Data: Addressing gaps in data is essential for maintaining dataset integrity, often handled by imputation or removal of incomplete entries.
Outliers
- Definition: An outlier is any data point more than \( 1.5 \times \text{IQR} \) away from the nearest quartile.
- Interpretation: Outliers may be valid parts of a sample or indicate errors. Context is crucial in determining their relevance.
Example:
- If the IQR (Interquartile Range) of a dataset is \( 15 \), any value more than \( 22.5 \) (\( 1.5 \times 15 \)) above the upper quartile or below the lower quartile is an outlier.
Visualization
Outliers are often identified using box-and-whisker diagrams, which visually represent data spread and anomalies. (Refer to SL4.2)
Sampling Techniques
Effective sampling methods minimize bias and improve representativeness. Common methods include:
- Simple Random Sampling: Each member of the population has an equal chance of selection.
- Convenience Sampling: Selection based on ease of access, though it may introduce bias.
- Systematic Sampling: Selecting every \( k \)-th individual from a list or sequence.
- Quota Sampling: Ensures certain groups are proportionally represented by setting quotas.
- Stratified Sampling: Divides the population into subgroups (strata) and samples from each strata proportionally.
Applications
- Market Research: Understanding consumer preferences through sampling.
- Healthcare: Studying treatment effects using stratified sampling techniques.
- Education: Evaluating performance across different student demographics.
Linkages
- Box-and-Whisker Diagrams: For visualizing data spread and detecting outliers. (SL4.2)
- Measures of Dispersion: Understanding data variability. (SL4.3)
IB Mathematics AA SL Concepts of population and sample Exam Style Worked Out Questions
Question
At a skiing competition the mean time of the first three skiers is 34.1 seconds. The time for the fourth skier is then recorded and the mean time of the first four skiers is 35.0 seconds. Find the time achieved by the fourth skier.
▶️Answer/Explanation
Markscheme
total time of first 3 skiers \( = 34.1 \times 3 = 102.3\) (M1)A1
total time of first 4 skiers \( = 35.0 \times 4 = 140.0\) A1
time taken by fourth skier \( = 140.0 – 102.3 = 37.7{\text{ (seconds)}}\) A1 [4 marks]
Question
Consider the data \(x_1,x_2,x_3, …, x_n,\) with mean \(\overline{x}\), and standard deviation \(s\).
- If each number is increased by \(k\),
- show that the new mean is \(\overline{x}+k\) (i.e. it is also increased by \(k\))
- show that the new standard deviation is \(s\)(i.e. it remains the same)
- If each number is multiplied by \(k\)
- show that the new mean is \(k\overline{x}\) (i.e. it is also multiplied by \(k\))
- show that the new standard deviation is \(ks\)(i.e. it is also multiplied by \(k\))
- write down the relation between the original and the new variance.
▶️Answer/Explanation
Ans:
- \(\overline{x}_{new}=\frac{\sum_{i=1}^{n}(x_i+k)}{n}=\frac{\sum_{i=1}^{n}x_i+\sum_{i=1}^{n}k}{n}=\frac{\sum_{i=1}^{n}x_i}{n}+\frac{\sum_{i=1}^{n}k}{n}=\overline{x}+\frac{kn}{n}=\overline{x}+k\)
\(s_{new}=\sqrt{\frac{\sum_{i=1}^{n} ((x_i+k)-(\overline{x}-k))^2 }{n}}=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\overline{x})^2)}{n}}=s\) - \(\overline{x}_{new}=\frac{\sum_{i=1}^{n}(kx_i)}{n}=\frac{k\sum_{i=1}^{n}(x_i)}{n}=k\frac{\sum_{i=1}^{n}(x_i)}{n}=k\overline{x}\)
\(s_{new}=\sqrt{\frac{\sum_{i=1}^{n}(kx_i-k\overline{x})^2}{n}}=\sqrt{\frac{k^2\sum_{i=1}^{n}(x_i-\overline{x})^2}{n}}=k\sqrt{\frac{\sum_{i=1}^{n}(x_i-\overline{x})^2}{n}}=ks\)
\(s_{new}=ks\)⇒\(s^2_{new}=k^2s^2\)
so the original variance is multiplied by \(k^2\).