Unit 8 · Statistics & ProbabilityS2 · Spread & Displays · Telemetry

Spread & Data Displays

Compare range, IQR, and standard deviation and read box, dot, and stem plots.

28 min
Unlocks: Signal Filters

Ready when you are

7 steps · about 28 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

range: Maximum minus minimum.
interquartile range: Q3 minus Q1; the spread of the middle half.
standard deviation: A measure of typical distance from the mean.

Start by naming the quantity each problem asks for. That habit blocks the most common ACT trap.

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Try it first

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Take a calm first shot. Label what you know, write one relationship, and make the best choice you can.

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The big idea

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Build the picture before the arithmetic

For standard deviation comparisons, you usually do not calculate. Look for values farther from their mean. A box plot encodes minimum, Q1, median, Q3, and maximum in that order.

Key idea

The decision that matters

Center tells where the data sit; spread tells how far they scatter. Moving every value equally changes center but not spread.

Interquartile range

\operatorname{IQR}=Q_3-Q_1

Write this relationship before inserting numbers. It keeps labels and units attached to the work.

A compact example

For $2,4,6,8,10$ , the median is 6, $Q_1=3$ , $Q_3=9$ , and $\operatorname{IQR}=9-3=6$ .

Cue	Best move
Middle-half spread	Subtract the first quartile from the third quartile
Overall dispersion	Compare distances from the mean

Trap

Confusing a higher mean with a larger spread

This error produces a believable answer. Pause before computing and say what each number means.

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Worked examples

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Skip check · 2 questions

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Worked example

Example 1: Find IQR

Problem

For the sorted data 2, 4, 5, 7, 9, 12, 15, 18, find the interquartile range.

The lower half is 2, 4, 5, 7, so

Q_1=(4+5)/2=4.5

The upper half is 9, 12, 15, 18, so $Q_3=(12+15)/2=13.5$ .

Which procedure finds the interquartile range?

IQR is $13.5-4.5=9$ .

Why are the quartiles found from the two halves?

Worked example

Example 2: Compare standard deviations

Problem

Set A is 48, 49, 50, 51, 52. Set B is 30, 40, 50, 60, 70. Which has the larger standard deviation?

Both sets have mean 50.

Set A stays within 2 of the mean; Set B reaches 20 away.

Which set has the larger standard deviation?

Set B has the larger standard deviation because its values are more dispersed.

Why does Set B have greater spread?

Faded example · you finish it

Example 3: You finish it

Problem

A box plot has

Q_1=14

and

Q_3=31

. Find the IQR.

Read

Q_1

and

Q_3

, then subtract

Q_3-Q_1

Which completion is correct?

Which reason validates the result?

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.