Data analysis

Idea

A list of numbers can be read in more than one way. A two-dimensional sample $\{(x_i,y_i)\}$ can be a cloud around its mean. Moving one point far away can change the trend. Keeping only one list of values turns the question into order.

This page does not derive each statistic. It sets three readings side by side: overall direction, the pull of one point, and ordered shape. Each mode keeps its own sample. In the cloud, the line is the least-squares line and $r$ is the Pearson correlation. The box uses the same inclusive percentile as the box-plot page. Its fences are $Q_1-1.5\mathrm{IQR}$ and $Q_3+1.5\mathrm{IQR}$, and the whiskers stop at the outermost values still inside them.

Interaction

• Scatter diagram and regression: drag the points, or set Point count n, Trend b, and Noise σ, and read the cloud’s direction and its mean
• Outlier influence: move only the highlighted point and see whether it rewrites the main trend. Outlier xₒ and Outlier yₒ follow that point
• Percentile box: switch to one-dimensional values. Sample size n rebuilds the list, and the median, the quartiles, and the tail outliers cut it
• Reset data rebuilds the current mode. Show guides and Hide guides toggle the guide lines. Add data and Delete selected change the cloud or the list; they are absent in Outlier influence. Delete selected removes the selected point only while more than three points remain in the cloud, or more than five in the list

Suggested order: Scatter diagram and regression, then Outlier influence, then Percentile box.

What to notice

• The cloud shows direction, the outlier comparison shows influence, and the box plot shows order.
• The mean and the least-squares line are easy for a high-influence point to pull. The median and the quartiles first ask where the middle of the ordered sample sits.
• An outlier is not one rule. In the plane, ask whether the point pulls the trend. On the number line, ask whether it falls past the whisker fences.

Related

• Scatter diagram, correlation, and the regression line
• An outlier’s effect on the regression line
• Percentiles and a box plot

Further reading

• Pearson correlation coefficient (Wikipedia)
• Simple linear regression (Wikipedia)
• Box plot (Wikipedia)
• Percentile (Wikipedia)