Fundamentals of Statistical Measurement and Data Analysis

Chapter 1: Understanding Variables

Types of Variables

• Categorical: Smoker (current, former, no)
• Ordinal: Non, light, moderate, heavy smoker (ordered categories)
• Quantitative: BMI, Age, Weight (numerical measurements)

Key Definitions

• Observation: Measurements are made (individual or aggregate).
• Variable: The generic characteristic we measure (e.g., age).
• Value: A realized measurement (e.g., 27).

Chapter 2: Statistical Studies

Surveys: Census and Sampling

• Goal: Describe population characteristics.
• Census: Attempts to reach the entire population (costly, time-consuming).
• Sampling: Uses a sample of the population (allows for inferences, saves time and money).
• Simple Random Sampling: Based on probability.
• Issues with Sampling: Under-coverage, volunteer bias, and nonresponse bias.

Comparative Studies: Experimental and Non-experimental

These studies determine the relationship between explanatory and response variables (e.g., a study on whether weight gain causes hypertension).

• Experimental: Subjects are assigned according to the explanatory variable (exposed/unexposed).
• Non-experimental: Subjects are not assigned; they are merely classified as exposed/nonexposed.

Blinding Techniques

• Single Blinding: Subjects are unaware of the specific treatment they receive.
• Double Blinding: Subjects and investigators are unaware of the treatment assignments.
• Triple Blinding: Subjects, investigators, and statisticians are unaware of the treatment assignments.

Chapter 3: Data Visualization and Distribution Shapes

Visualizing Data

Stem Plot: Rotated:

Distributional Shapes

• Modality: The number of peaks in the distribution.
• Kurtosis: The steepness or peakedness of the distribution.

Frequency Tables and Charts

Frequency Table

• Histogram: Used for displaying quantitative measurements.
• Bar Chart: Used for displaying categorical measurements.

Chapter 4: Measures of Central Location and Spread

Descriptive Measures

• Central Location: Mean, Median, Mode
• Spread: Range, IQR, Variance, and Standard Deviation

Notation

• n = Sample size
• N = Population size
• X = Variable (e.g., ages of subjects)
• x_i = The value of the individual i for X
• Σ = Sum of all values (Capital Sigma)

Measures of Central Location

Mean: The Arithmetic Average ("x-bar")

The mean is the balancing point of a distribution.

Median: The Middle Value

The median is more robust in the face of outliers or errors.

Relationship Between Mean and Median (Skew)

• Symmetric Data Set: Mean = Median = Mode
• Symmetrical Distribution: Mean = Median
• Positive Skew (Right Skew): Mean > Median
• Negative Skew (Left Skew): Mean < Median

Measures of Spread

Range

Range = Maximum value - Minimum value. (Note: Sample range tends to underestimate population range.)

Quartiles and IQR

• Q1 (First Quartile): Cuts off the bottom quarter of data; it is the median of the lower half of the data set.
• Q3 (Third Quartile): Cuts off the top quarter of data; it is the median of the upper half.
• IQR (Interquartile Range): Q3 - Q1. Covers the middle 50% of the distribution.

**When n is odd, include the median in both halves of the data set when calculating quartiles.**

Boxplot

Standard Deviation

The most common descriptive measure of spread.

Rules and Reporting

Chebychev's Rule (Applicable to all distributions)

At least 75% of the values fall within the range μ ± 2σ.

Rounding

Carry at least four significant digits during calculations.

Choosing Summary Statistics

Always report measures of central location, spread, and sample size.

• For symmetrical, mound-shaped distributions: Report Mean and Standard Deviation.
• For skewed or odd-shaped distributions: Report 5-point summaries (Median and IQR).