Exploratory Data Analysis for Univariate Quantitative Data

Statistical Analysis Process and Variable Types

• Process of Statistical Analysis:

  • Identify the population and the parameter of interest ($\mu$).

  • Collect sample data.

  • Posit a statistical model based on sample information.

  • Draw inference about the population parameter using the model.

  • Note: Sample statistics ($\bar{y}$) are related to population parameters ($\mu$), but $\bar{y} \neq \mu$.

• Types of Variables:

  • Continuous Quantitative: Can take any numerical value, including decimals (e.g., percentage of bone density loss).

  • Discrete Quantitative: Can take only specific, fixed numerical values (e.g., count of bedrooms).

  • Categorical: Takes values from a fixed set of non-numerical categories (e.g., college of major).

Graphical Tools for Exploratory Data Analysis

• Histogram:

  • Groups quantitative data into specified bin ranges and plots counts per bin.

  • Always described using Shape, Center, and Spread.

  • Sensitive to bin size selection.

• Density Plot:

  • A smooth, continuous representation of a histogram.

  • Total area under the density curve is exactly $1$.

• Boxplot:

  • A visual plot of the 5-number summary ($Min$, $Q_1$, $M$, $Q_3$, $Max$).

Numerical Summaries

• Measures of Center:

  • Mean ($\bar{y}$): The balancing point of the histogram:     $\bar{y} = \frac{1}{n}\sum_{i=1}^{n} y_i$

  • Median ($M$): The middle value dividing the data such that $50\%$ lies below and $50\%$ lies above.

  • Mode: The peak location corresponding to the most frequent value.

• Measures of Spread:

  • Standard Deviation ($S$): Represents the average distance of data points from the mean:     $S = \sqrt{\frac{\sum_{i=1}^{n} (y_i - \bar{y})^2}{n - 1}}$

  • Interquartile Range ($IQR$): Measures the spread of the middle $50\%$ of observations:     $IQR = Q_3 - Q_1$

  • Range: Distance between extremes ($Max - Min$).

  • Properties: Spread is always $\ge 0$, equals $0$ only when all values are identical, and shares the same units as the original data.

• Measures of Shape:

  • Skewness: Quantifies symmetry within range $-\infty < \text{skew} < \infty$.

  • $\text{skew} < 0$: Left-skewed distribution.

  • $\text{skew} > 0$: Right-skewed distribution.

  • $-0.5 < \text{skew} < 0.5$: Approximately symmetric distribution.

Outliers and Selection of Descriptive Statistics

• Outlier Detection ($1.5 \times IQR$ Rule):

  • An observation is an outlier if:     $\text{Value} > Q_3 + 1.5 \times IQR$     or     $\text{Value} < Q_1 - 1.5 \times IQR$

• Selecting Summary Measures:

  • Mean chases outliers and Standard Deviation becomes inflated by extreme values.

  • Presence of outliers or strong skewness: Use Median and IQR.

  • Absence of outliers and strong skewness: Use Mean and Standard Deviation.