Distributions & Summary Statistics

Graphical Displays

• Histogram: shows frequency distribution; good for shape.
• Stemplot: shows individual values; good for small datasets.
• Boxplot (Box-and-Whisker): shows five-number summary; good for comparing groups.
• Dotplot: shows each data value as a dot.

Describing Shape

• Symmetric: roughly equal on both sides of center.
• Skewed right: tail extends to the right (positive skew).
• Skewed left: tail extends to the left (negative skew).
• Unimodal / Bimodal / Uniform.

Measures of Center

• Mean ($\bar{x}$): $\bar{x} = \frac{\sum x_i}{n}$ — sensitive to outliers.
• Median: middle value — resistant to outliers.
• For skewed data, median is preferred.

Measures of Spread

• Range: max − min.
• Interquartile Range (IQR): $Q_3 - Q_1$ — resistant to outliers.
• Standard deviation ($s$): average distance from the mean.
• Variance ($s^2$): $s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}$.

Five-Number Summary

Min, $Q_1$, Median, $Q_3$, Max.

Outliers (1.5×IQR Rule)

A value is a potential outlier if: $x < Q_1 - 1.5 \cdot IQR \quad \text{or} \quad x > Q_3 + 1.5 \cdot IQR$

Effects of Transformations

• Adding a constant $c$: center shifts by $c$; spread unchanged.
• Multiplying by $c$: center and spread both multiply by $|c|$.

Dataset: 12, 15, 17, 19, 20, 22, 24, 28, 35

$\bar{x} = 192/9 = 21.3$. Median = 20 (5th value). $Q_1 = 16$, $Q_3 = 26$. $IQR = 10$. Outlier fence: $Q_1 - 15 = 1$, $Q_3 + 15 = 41$. No outliers.

• Always describe shape, center, spread, and outliers.
• Mean and standard deviation for symmetric data; median and IQR for skewed data.
• 1.5×IQR rule identifies potential outliers.
• Transformations: adding shifts center; multiplying scales everything.