Lecture Notes on Measures of Spread in Data

Introduction

Purpose: Understand how to use numbers to describe the spread of data.
Variability: Measure of differences within data samples.
- Low variability: data points are similar.
- High variability: data points are different.

Maximum and Minimum
- Easiest measure of spread.
- Represents the range of data values.
- Sensitive to outliers (e.g., a single large outlier can shift max/min significantly).
Quartiles
- Median (Q2): Middle value of the ordered data.
- First Quartile (Q1): Value between the lowest value and the median.
- Third Quartile (Q3): Value between the median and the highest value.
- Quartiles are robust to outliers.
- Data split into quarters, each containing 25% of the data.
Variance
- Measures the mean of squared deviations from the mean.
- Population variance uses 'N'; sample variance uses 'N-1' (degrees of freedom).
  - N-1: Corrects for statistical bias when estimating a population from a sample.
- Provides an average squared deviation, which is less intuitive.
Standard Deviation (SD)
- Square root of variance.
- Easier to interpret as average deviation from the mean.
- Most data falls within one SD from the mean, outliers are beyond two or three SDs.

1.5 IQR Rule: Identifies potential outliers.
- Calculate IQR (Q3 - Q1), multiply by 1.5.
- Maximum cutoff: Q3 + (1.5 * IQR).
- Minimum cutoff: Q1 - (1.5 * IQR).
- Values outside these cutoffs are potential outliers.

Possible Causes:
- Typographical errors.
- Measurement errors.
- Incorrect identification of the sample.
- Legitimate extreme values.
Deciding on Outliers:
- Inclusion: Considered valid data within population.
- Exclusion: If the data point is not representative or caused by error.
- Decision should be case-specific and explained.