6. Data Analysis: Statistics and Box Plots

Measures of center:

• Mean: sum of all values ÷ count.
• Median: middle value (or average of two middle values) when data is ordered.
• Mode: the most frequent value.

Measures of spread:

• Range: max − min.
• Interquartile range (IQR): Q3 − Q1 (middle 50% of data).
• Outlier rule: a value is an outlier if it falls more than \(1.5 \times IQR\) below Q1 or above Q3.

Box plot (5-number summary): minimum, Q1, median, Q3, maximum.

Effect of adding a constant or outlier:

• Adding a constant to every value shifts the mean by that constant; spread stays the same.
• An outlier pulls the mean toward it but does not change the median much.

Weighted mean: multiply each value by its weight, sum, then divide by total weight.
\[ \bar{x} = \frac{w_1 x_1 + w_2 x_2 + \cdots}{w_1 + w_2 + \cdots}. \]

Scatter plots: direction (positive/negative correlation), form (linear/nonlinear), strength (tight vs. loose cluster). The line of best fit (trend line) models the relationship.