🌊Range & Spread

Two bus routes both claim "5 minutes late on average". Sounds like a tie? In truth one runs four or five minutes late every day, while the other is usually on time but now and then slips by half an hour at a stretch. The summaries from Mean, Median & Mode can't see this difference at all — we need one more quantity to say how far apart the data spreads.

One average, two personalities

• Group A: $4, 5, 5, 6$ — mean 5, every value huddled around 5;
• Group B: $0, 0, 0, 20$ — mean also 5, yet swinging wildly, propped up by a single 20.

Same average, totally different temperament. The quantity describing that temperament is called spread, and the simplest ruler for it is the range.

Range: maximum minus minimum

Range = maximum − minimum.

Group A: $6 - 4 = 2$; Group B: $20 - 0 = 20$. Group B's range is ten times Group A's, and the restlessness shows at once. The range captures the two farthest endpoints in one sweep — quick, but it uses only two numbers, so whether the middle data huddles or scatters, it has no idea.

A finer ruler: quartiles

Line the data up from smallest to largest and cut it into four equal parts. Take $2, 4, 6, 8, 10, 12, 14$: the middle value $8$ is the median; the middle of the lower half, $4$, is called Q1, and the middle of the upper half, $12$, is Q3. The middle half of the data lives between $Q1$ and $Q3$, and that distance

$IQR = Q3 - Q1 = 12 - 4 = 8$

is the interquartile range. It only watches the middle 50% of the data, so no matter how the endpoints thrash about, it barely blinks.

Outliers: champions at stretching the range

Slip a $98$ into $2, 3, 3, 4$: the range leaps from 3 to $98 - 2 = 96$, yet the rest of the data still huddles between 2 and 4. Such a towering stranger is an outlier — the moment it arrives, the range stops telling the truth. The IQR is more trustworthy here: 98 sits outside the middle 50%, so the IQR hardly moves.

This is also the referee for the "bus vs train" debate: when the average delay is the same, compare ranges or IQRs and you see instantly who is steady and who is erratic — and which monthly pass to buy.

Spread Lab

+🗑 Clear4865389

Count

7

Mean

6.14

Median

6

Mode

8

Edit the numbers freely and watch the bars scatter and squeeze while the mean line chases along. Two experiments: drag one number very large — the mean gets dragged away while the median stands its ground, the raw power of an outlier; then squeeze all the numbers together — the summaries barely change, but "scattered" versus "tight" is plain to the eye. Compute the range yourself: maximum minus minimum, right there in the numbers you just edited.

Check yourself

✅

Quick quiz

↺ Try again

1. 1. What is the range of 3, 7, 2, 9, 4?

   795

2. 2. Two classes both average 70. Class A scores between 65 and 75, class B between 40 and 100. Which is more spread out?

   Class BClass Aequally spread

3. 3. Add 100 to 2, 3, 4, 5 (range 3). Which gets dragged harder, the range or the IQR?

   the rangethe IQRneither changes