Summary of Fundamental Structure of Matter

## Introduction Biostatistics applies statistical methods to questions in biology, medicine and public health. It helps summarize data, assess variability, test hypotheses about treatments, and draw conclusions from samples so we can make informed decisions about patient care, clinical trials, and risk assessment. > Definition: Biostatistics is the field that uses statistical reasoning and methods to analyze biological and medical data, design studies, and interpret results. ## 1. Types of data Break data into two main groups and their subtypes. Clear classification helps choose the right summary statistics and tests. ### Qualitative (categorical) data - **Nominal:** categories with no natural order, e.g., eye colour: blue, green, brown. - **Ordinal:** categories with a meaningful order, e.g., disease state: mild, moderate, severe; education level: elementary, secondary. > Definition: Qualitative data describe attributes or categories and are not numerical. ### Quantitative (numerical) data - **Continuous:** values can take any number in an interval, e.g., height, blood glucose level. - **Discrete:** values take specific counts, e.g., number of hospital visits. > Definition: Quantitative data are numeric measurements that permit arithmetic operations. Table: Qualitative vs Quantitative | Feature | Qualitative | Quantitative | | --- | ---:| ---:| | Nature | Categories | Numbers | | Examples | Eye colour, disease stage | Height, weight, counts | | Common summaries | Mode, proportions | Mean, median, SD | Fun fact: Choosing the wrong data type for analysis can lead to meaningless results, for example treating ordinal pain scores as interval data without justification. ## 2. Descriptive vs Inferential statistics - **Descriptive statistics:** summarize and present observed data (averages, tables, graphs). Useful for reporting study results. - **Inferential statistics:** draw conclusions about a larger population from a sample using tests, confidence intervals, and models. > Definition: Descriptive statistics summarize observed data; inferential statistics estimate and test hypotheses about populations using sample data. Practical application: In a clinical trial you first describe baseline characteristics, then use inferential tests to decide whether a drug effect likely generalizes beyond the study sample. ## 3. Measures of location (central tendency) - **Mean (arithmetic average):** sum of values divided by count: $$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$$ - **Median:** middle value when data are ordered; robust to outliers. - **Mode:** most frequent value, useful for categorical data. When to use: mean for symmetric interval data without extreme outliers; median for skewed distributions; mode for categorical data. ## 4. Measures of variability (spread) - **Range:** $$R = x_{\text{max}} - x_{\text{min}}$$ - **Variance (sample):** $$s_x^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n-1}$$ - **Standard deviation (sample):** $$s_x = \sqrt{s_x^2}$$ - **Standard error of the mean (SEM):** $$se_x = \frac{s_x}{\sqrt{n}}$$ - **Coefficient of variation:** $$V_x = \frac{s_x}{\bar{x}} \cdot 100\%$$ Practical example: Compare two treatment groups' mean blood pressure and their SDs to assess both central effect and consistency. Did you know that the SEM decreases as sample size increases because $$se_x = s_x / \sqrt{n}$$, so larger samples give more precise estimates of the mean? ## 5. Uncertainty and reporting - Always report a measured value with its uncertainty: $x_0 \pm \Delta x$ where $\Delta x$ is absolute uncertainty. - Relative uncertainty: $$\Delta_r x = \frac{\Delta x}{x_0}$$ - Percentage uncertainty: $$\Delta_{\%} x = \frac{\Delta x}{x_0} \cdot 100\%$$ Rounding rules (practical): - Keep calculations precise; round only final results. - Express uncertainty with up to two significant digits; round the main value to the same decimal place as the uncertainty. Example: If mean height is $167.68$ cm and SD is $5.27$ cm,