The middle half and the 1.5·IQR rule

Cutting the sorted data into quarters

Sort the twelve amounts and cut them into four equal-count pieces. The cut points are the quartiles: Q1 has a quarter of the values below it, Q2 is the median, Q3 has three quarters below it. The interquartile range, IQR = Q3 − Q1, is the width of the middle half of the data — the rank-based spread that pairs with the median. A percentile generalises the idea: the 90th percentile has 90 % of the values at or below it.

Worked example — the amounts

With $D$2:$D$13 as the range:

• =QUARTILE.INC($D$2:$D$13, 1) → 241.25
• =QUARTILE.INC($D$2:$D$13, 3) → 670
• IQR = 670 − 241.25 = 428.75
• =PERCENTILE.INC($D$2:$D$13, 0.9) → 958

The fractional Q1 is not a mistake: with twelve values the quarter mark falls between the 3rd (215) and 4th (250) sorted values, and the function interpolates. R's quantile() does the same by default:

> quantile(orders$amount, c(0.25, 0.5, 0.75))
   25%    50%    75% 
241.25 445.00 670.00 
> IQR(orders$amount)
[1] 428.75

The five-number summary — minimum, Q1, median, Q3, maximum — is 90 · 241.25 · 445 · 670 · 1320, and it is what a box plot draws: the box spans Q1 to Q3, the line inside is the median, the whiskers reach to the last values inside the fences. R's summary(orders$amount) prints the same five plus the mean, rounded to four significant digits (1st Qu. 241.2). One R habit to unlearn: range(orders$amount) returns two numbers, [1] 90 1320, the minimum and maximum — not their difference. For the single-number range use diff(range(x)) or MAX − MIN.

The 1.5·IQR fences

An outlier is a value that sits far outside the middle half. The standard rule fences it:

$$\text{lower fence}=Q1-1.5\cdot IQR,\text{upper fence}=Q3+1.5\cdot IQR$$

Here 241.25 − 643.125 = −401.88 and 670 + 643.125 = 1313.13. No amount can be negative, so the lower fence is moot; the upper fence catches exactly one order — 1005, at $1,320. What a fenced point means is a judgement, not a verdict: it is flagged for investigation. Is the $1,320 a typo for $132? A genuine bulk purchase? A different kind of customer? The analyst finds out. A fenced value is never deleted automatically and never replaced by the median; both moves destroy information and, in a twelve-row dataset, would erase Alberta's largest sale.

Type four calls, then fill the worksheet

Type the two spreadsheet functions and the two R calls. The worksheet asks for Q1, Q3, the IQR and the upper fence.

CRISP-DM: quartiles and fences are data understanding → explore data; investigating the flagged order is verify data quality.