Catching Glitches in a Distance Sensor

A robot's ultrasonic sensor reads the distance to a wall twenty times. Most readings agree, but now and then an echo bounces off something else and reports nonsense. A box plot's fences flag those glitches automatically.

How it works

Quartiles, box plots and the 1.5 × IQR rule

In one sentence

The quartiles $Q_1$ and $Q_3$ are the 25th and 75th percentiles, and $\text{IQR} = Q_3 - Q_1$ . Tukey's rule flags values below $Q_1 - 1.5\,\text{IQR}$ or above $Q_3 + 1.5\,\text{IQR}$ as outliers; a box plot draws the box from $Q_1$ to $Q_3$ , a line at the median and whiskers to the most extreme values inside the fences.

General formula

{\htmlData{sym=0}{\text{IQR}}} = {\htmlData{sym=2}{Q_3}} - {\htmlData{sym=1}{Q_1}}, \qquad \text{outlier if } {\htmlData{sym=4}{x}} < {\htmlData{sym=1}{Q_1}} - {\htmlData{sym=3}{k}}\,{\htmlData{sym=0}{\text{IQR}}} \;\text{ or }\; {\htmlData{sym=4}{x}} > {\htmlData{sym=2}{Q_3}} + {\htmlData{sym=3}{k}}\,{\htmlData{sym=0}{\text{IQR}}}

The box is where the typical readings live. Anything farther than one and a half box-widths away is suspicious.

What the symbols mean

$\text{IQR}$
Interquartile range
The width of the box: the spread of the middle half of the readings.
$Q_1$
First quartile
A quarter of the readings are at or below it.
$Q_3$
Third quartile
Three quarters of the readings are at or below it.
$k$
Fence multiplier
How many IQRs beyond the box a reading may sit before it's flagged; 1.5 is the usual choice.
$x$
A reading
One distance measurement, in centimeters.

Sort the readings. The quartiles $Q_1$ and $Q_3$ cut off the lowest and highest quarter, so the box spans the middle half and its width is the IQR.
Fences sit $k$ IQRs beyond the box, with $k = 1.5$ by convention. A reading outside a fence is flagged as an outlier.
The whiskers reach the most extreme readings still inside the fences.
Because quartiles ignore the extremes, a glitch can't stretch the fences to hide itself, unlike a rule based on the mean and standard deviation.

Learn the math

Quartiles, Box Plots and OutliersStats · Lesson 8, Descriptive Statistics Summarizing Process DataEng Stats · Lesson 2, Engineering Data and Variation Prediction IntervalsEng Stats · Lesson 14, Intervals for Engineering Decisions Residuals and DiagnosticsEng Stats · Lesson 21, Regression for Engineering

Learn the math

Quartiles, Box Plots and OutliersStats · Lesson 8, Descriptive Statistics

Summarizing Process DataEng Stats · Lesson 2, Engineering Data and Variation

Prediction IntervalsEng Stats · Lesson 14, Intervals for Engineering Decisions

Residuals and DiagnosticsEng Stats · Lesson 21, Regression for Engineering