Shape, Outliers, and Reporting an Answer Honestly

In this lesson you will: Describe a distribution's shape: symmetric, skewed left, skewed right, clustered or gapped; Test a suspected outlier with the 1.5rule and report its effect on center and spread; Choose a level of accuracy that the measurements actually justify.

Shape is a description, and it has vocabulary

Before any statistic gets computed, look at the picture and say what it looks like. A symmetric distribution has a mirror-like balance around its center. A skewed right distribution has a long tail toward the high values, which is what money, waiting times and text-message counts almost always do. Skewed left is the mirror image and is rarer.

Two more words earn their place. A cluster is a group of values packed together; a gap is a stretch of the scale where nothing landed at all. Neither is an error. Two clusters with a gap between them usually means two different kinds of thing got measured and put in the same list.

Shape decides which summary you report. Symmetric means mean and standard deviation; skewed means median and IQR.

An outlier is decided by a rule, not by eye

The largest value in a data set is not automatically an outlier. Some data sets have no outliers at all, and in a large data set a value can look extreme and still be ordinary for that distribution.

The standard test uses the quartiles. Compute the IQR, then build two fences. Anything outside them is an outlier.

Report an answer the measurements can support

A calculator will hand you 33.583333. Almost none of those digits mean anything. N.Q.3 asks you to choose accuracy that matches the limits of the measurement, and there are only a few rules to carry.

Round once, at the very end. Rounding partway through and then continuing to compute pushes the error forward into every line that follows.

Common mistake — Calling the biggest value an outlier

Handed the list 12, 14, 15, 16, 18, 19, 21, 24 and asked to identify outliers, a student points at 24.

The instinct is reasonable: 24 is the largest, it is six away from its neighbor 18 at a glance, and the question seemed to promise that there was one to find.

Run the rule instead. The median is (16+18)/(2)=17; Q_1=(14+15)/(2)=14.5 and Q_3=(19+21)/(2)=20. The IQR is 5.5, so the upper fence is 20+1.5(5.5)=28.25. The value 24 is comfortably inside it.

This data set has no outliers. \"Largest\" and \"outlier\" are different words, and a question can legitimately have the answer \"none.\"

How to catch it: Write the fence down as a number before you name any value. If you cannot state the number a value has to beat, you are guessing rather than testing.