Chapter 11: Biostatistics and Data Analysis

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Chapter Overview & SLOs

What is biostatistics? Biostatistics is the application of statistical methods to biological data. It helps in collecting, organizing, analyzing, and interpreting data to draw meaningful conclusions in biology and medicine.

What is the arithmetic mean? The mean is the sum of all values in a dataset divided by the total number of values.

• Formula for ungrouped data: $\bar{x} = \frac{\sum x}{n}$
• Where: $\sum x$ = sum of all values, $n$ = total number of values
• Symbol: $\bar{x}$ (read as "x-bar")
• Important: The mean is highly sensitive to outliers (extreme values), which can significantly skew the true central tendency.

How is the median determined? The median is the middle value in a dataset arranged in ascending or descending order.

• Symbol: $\tilde{x}$ (read as "x-tilde")
• For odd number of values ($n$ odd): Median = $(\frac{n+1}{2})^{th}$ term
• For even number of values ($n$ even): Median = $\frac{(\frac{n}{2})^{th} \text{ term} + (\frac{n}{2} + 1)^{th} \text{ term}}{2}$
• Median is not affected by outliers.

How is the mode determined? The mode is the value that appears the maximum number of times in a dataset.

• Symbol: $\hat{x}$ (read as "x-hat")
• A dataset can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes).
• If all values appear the same number of times, there is no mode.

How do we estimate the mode from a continuous frequency distribution? For grouped data, we use the interpolation formula:

• $\hat{x} = l + \frac{(f_m - f_1) \times h}{(f_m - f_1) + (f_m - f_2)}$
• Where:
  • $l$ = lower boundary of the modal class (class with highest frequency)
  • $h$ = class width
  • $f_m$ = frequency of the modal class (maximum frequency)
  • $f_1$ = frequency of the class before the modal class
  • $f_2$ = frequency of the class after the modal class

What are frequency distributions and cumulative frequency?

• Frequency distribution: A table that organizes data into classes (intervals) and shows the number of observations (frequency) in each class.
• Cumulative frequency: The running total of frequencies. It shows the number of observations less than or equal to the upper boundary of each class.

How do we differentiate between bar charts and pie charts?

| Feature | Bar Chart | Pie Chart | |---------|-----------|-----------| | Purpose | Compare quantities across categories | Show proportions of a whole | | Axes | X and Y axes (categories vs values) | No axes | | Shape | Rectangular bars | Circular sectors (slices) | | Best for | Comparing multiple categories | Showing percentage distribution | | Example | Number of students in different grades | Budget allocation percentages |

What is the effect of outliers on mean? Outliers are extreme values that are very different from the rest of the data. They can significantly affect the mean, pulling it toward the outlier. For this reason, median is often preferred when outliers are present (e.g., income data, test scores).

Summary of central tendency measures:

| Measure | Symbol | Formula | Sensitivity to Outliers | |---------|--------|---------|------------------------| | Mean | $\bar{x}$ | $\frac{\sum x}{n}$ | High | | Median | $\tilde{x}$ | Middle value | Low | | Mode | $\hat{x}$ | Most frequent value | Low |

These notes are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do we calculate the Mean for ungrouped data? We use the formula $\bar{x} = \frac{\sum x}{n}$, where $\sum x$ is the sum of all values and $n$ is the total number of items. The mean is sensitive to outliers.
• How do we find the Median for an even-numbered dataset? We arrange the data in order, identify the two middle terms ($(n/2)^{th}$ and $(n/2 + 1)^{th}$ terms), and calculate their average: $\tilde{x} = \frac{(n/2)^{th} \text{ term} + (n/2 + 1)^{th} \text{ term}}{2}$.
• How do we estimate the Mode from a continuous frequency distribution? We apply the standard interpolation formula: $\hat{x} = l + \frac{(f_m - f_1) \times h}{(f_m - f_1) + (f_m - f_2)}$, where $f_m$ is the maximum frequency, $l$ is the lower boundary of the modal class, $h$ is class width, $f_1$ is frequency before modal class, and $f_2$ is frequency after modal class.
• How do we differentiate between Bar Charts and Pie Charts? Bar Charts use X and Y axes to compare quantities across categories, while Pie Charts use circular sectors to visualize proportions of a whole.

Frequently Asked Questions (FAQ)