Choosing the Right Summary

The earlier lessons in this module handed you a whole toolbox. You can measure the center of a dataset with the mean, the median, or the mode. You can measure its spread with the range, the standard deviation, or the IQR. And you can show its shape with a histogram or a box plot.

This capstone adds no new formulas. Instead it answers the question that trips people up on real data: given a particular dataset, which of those tools should you actually report? Reach for the wrong summary and you can paint a misleading picture without ever making an arithmetic mistake.

The reason is resistance — how much a summary moves when one value goes to an extreme.

The mean is a balance point, so it feels every value's exact size. In the mean vs. median lesson you saw a single dragged outlier make the mean lurch across the number line while the median barely twitched. The standard deviation is even more sensitive: it squares each distance from the mean, so one far-away point dominates it. Mean and SD are not resistant.

The median is just the middle value by position, and the IQR is the width of the middle 50%. Neither cares how far away the extreme points sit — only that they are above or below. Median and IQR are resistant. That is exactly why they are the honest choice when the data is lopsided: they report what a typical value looks like instead of a number inflated by a handful of extremes.

A box plot is built entirely from resistant, position-based numbers — the five-number summary — so it is the perfect place to see shape and spot trouble. Two quick tells:

• Skew: if the median line sits off-center inside the box, or one whisker is much longer than the other, the data is skewed toward the long side.
• Outliers: any point drawn as a separate dot beyond the whiskers falls outside the 1.5×IQR fences — a flag that the mean and SD would be distorted.

In the tool below, switch the Data shape between symmetric, right-skewed, and with-an-outlier, and hit Regenerate a few times. Watch the box slide off-center for skewed data and a red dot break away when there is an outlier — the exact situations where you would abandon the mean and SD for the median and IQR.

Describing one dataset is half the job; often you need to compare several. Drawing side-by-side box plots on a shared axis is the cleanest way to do it, because each box shows a group's center, spread, and outliers in the same picture.

Scanning across them, you can read at a glance which group has the higher median (compare the middle lines), which is more spread out (compare box widths and whisker lengths), and which has unusual values (look for stray dots) — no new formula required. When groups are skewed or have outliers, comparing box plots is far more honest than comparing a single mean per group, which could hide a lopsided distribution behind one number.