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S01.5 · Relationships between variables

A free, illustrated module of 4 lessons and one interactive in S01 Exploratory Data Analysis: Scatter plots and covariance; Correlation; What a correlation hides: Anscombe’s quartet; Comparing groups: when one variable is a category.

S01.5 L1 · Core 4 lessons · about 100 min Live · in review

Part of S01 Exploratory Data Analysis in the Statistics and Data Analysis track.

Most questions about a table are about two columns at once: does rent rise with area, does a balcony depend on the district? These four lessons read a scatter plot, measure how two numbers move together with the covariance and the correlation, show with Anscombe’s quartet what a correlation cannot see, and compare groups when one column is a category.

What you need first. The mean, the standard deviation and z-scores (S01.3), and the boxplot (S01.4). Sum notation (M01.7) is used throughout.

Lessons

1. Scatter plots and covariance S01.5.1 25 min

   Plot two columns against each other and read the cloud for direction, form, strength and unusual points. Then measure how they move together with the covariance: a sum of signed rectangles, worked by hand on five points and on the nine flats.

2. Correlation S01.5.2 25 min

   Divide the covariance by both standard deviations and you get Pearson’s r: a number with no units that lies between -1 and 1. Learn it as the average product of z-scores, prove its range, and meet the trend line whose slope it sets.

3. What a correlation hides: Anscombe’s quartet S01.5.3 25 min

   Four sets of eleven points share their means, variances, correlation 0.816 and trend line, yet one is a cloud, one a curve, one a line with an outlier and one a stack. Learn why r sees only straight lines, how one point can make it, and how Spearman’s rank correlation resists.

4. Comparing groups: when one variable is a category S01.5.4 25 min

   When one column is a category, compare the groups side by side: their sizes, means, medians and boxplots. When both are categories, tally a two-way table and read its joint, marginal and conditional proportions, comparing rows to see whether the columns go together.

The interactive

S01.5-X1 · Anscombe explorer. Load any of Anscombe’s four sets of eleven points, then drag, move or remove a point. The means, standard deviations, Pearson’s r, Spearman’s rank correlation and the trend line update, and a switch hides the plot to show how little the table alone reveals. It lives in What a correlation hides: Anscombe’s quartet.