Week 02 · Outliers and the Normal Model

Monday

One lap far from the rest

// One far lap, and a bell-shaped hill

⏱ about 20 min

Monday: One Lap Far From the Rest

The crew's table sits under the big tree by the loop. Wren spreads out Wednesday's lap sheet.

"Twelve laps," he says. "52, 54, 55, 55, 56, 56, 56, 57, 57, 58, 60, 72. What do you notice?"

"That 72," Comet says. "I stopped to tie my shoe. It sits way off by itself."

Wren circles the far dot. "One value like that can pull the mean. Does it pull the median too?"

Nova shows both centers. "Mean 57.3. Median 56. Would you like a hint? Try the set without that lap."

"Without it, the mean drops to 56," Comet says. "The median stays at 56. It did not budge."

"We need a rule for which values count as far off," Wren says. "Not a feeling. A rule with a number."

"What can we make?" Comet asks. "A fence. Anything outside the fence is an outlier."

Wren circles one far dot on a dot plot while Comet holds a bell-shaped sketch beside stretch-timing cards.

Comet's Wednesday laps

Every number this week is the crew's own made-up reading from Nova's Run Log. Twelve laps, and one of them includes a shoe stop.

Lap	1	2	3	4	5	6	7	8	9	10	11	12
Seconds	52	54	55	55	56	56	56	57	57	58	60	72

$A dot plot of Comet's laps from 52 to 72 seconds, with one dot far to the right at 72.$

A solved problem to study: the 1.5 × IQR rule

Here is how Wren decides whether the far lap is an outlier. The rule uses the quartiles, so it does not depend on the far value itself.

Find the quartiles: Q1 = 55 and Q3 = 57.5. The IQR is 57.5 - 55 = 2.5.
Multiply the IQR by 1.5: 1.5 × 2.5 = 3.75.
Upper fence: Q3 + 1.5 × IQR = 57.5 + 3.75 = 61.25.
Lower fence: Q1 - 1.5 × IQR = 55 - 3.75 = 51.25.
Any value outside the fences is an outlier. 72 is above 61.25, so it is one. 60 is inside, so it is not.

$A box plot of Comet's laps with a long right whisker out to 72.$

What the outlier does

Measure	With the far lap	Without it
Mean	57.3	56
Median	56	56
Standard deviation	4.8	2
IQR	2.5	2

The mean moves by 1.3 seconds and the standard deviation more than doubles. The median does not move at all.

Measures that barely move are called resistant. The median and IQR are resistant. The mean and standard deviation are not.

FIND THE FENCES

Read the question.
Tap your answer.

Comet's laps have quartiles 55 and 57.5. What is the IQR?

Comet's laps have Q3 = 57.5 and IQR 2.5. Using the 1.5 × IQR rule, what is the upper fence?

Comet's laps are 52, 54, 55, 55, 56, 56, 56, 57, 57, 58, 60, 72. Which value is an outlier by the 1.5 × IQR rule?

Comet's laps are 52, 54, 55, 55, 56, 56, 56, 57, 57, 58, 60, 72. Remove the outlier 72. What is the mean now? (Round to 1 place.)

Comet's median lap was 56 with the far lap. What is the median without it? Type the number.

WHY THIS EXERCISEThe median did not move. That is what resistant means.

Statement	True or false?
57.5 + 1.5 × 2.5 = 61.25	?
An outlier pulls the mean toward itself.	?
The median moves as far as the mean when one value becomes extreme.	?
A lap of 60 seconds is an outlier in Comet's set.	?

WHY THIS EXERCISEThe rule replaces a feeling with a number, so two people reach the same answer.

Draw Comet's box plot twice: once with the far lap and once without it. Label both medians.

Run Log: copy the twelve laps, the fences and the outlier, and write the mean and median with and without it.Write one sentence about why the shoe stop should be reported separately rather than mixed in.Look back at your Track Day data from last week. Check it against the 1.5 × IQR rule.

A fence with a number. Tomorrow the stretch cards come out, and a very different shape appears: a bell.

Grown-up checked my answers

Tuesday →