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All courses Math I · S-ID.3 52 of 59
Interpret differences in shape, center, spread, and outliers in context.

Classify the shape of a dot plot

Problem
Classify the dot plot with counts 1:1,2:2,3:4,4:2,5:1. Report its modality, symmetry or skew, gaps, visual outliers, and supporting frequency evidence.
Source statistical display: Which complete classification correctly describes a dot plot with counts {1:1,2:2,3:4,4:2,5:1}? No requested comparison or classification is annotated. Open full size
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Hint

Locate maximum counts, compare equal-distance pairs around a center, then scan zero-frequency ticks and test any isolated extreme.

A distribution can have several simultaneous shape properties: modality, symmetry/skew, uniformity, gaps, and outliers. Report them in separate fields.

Solution walkthrough

01

Locate the modal value

\[\text{counts}~1,2,4,2,1~\text{have}~\text{unique}~\text{maximum}~4~\text{at}~\text{value}~3\]

One unique tallest stack makes the distribution unimodal with mode 3.

02

Test symmetry around 3

\[\text{count}~\text{at}~1=\text{count}~\text{at}~5=1;~\text{count}~\text{at}~2=\text{count}~\text{at}~4=2\]

Values equally far from 3 have equal frequencies, forming a balanced mound around the center.

03

Check for gaps

\[\text{every}~\text{integer}~\text{from}~1~\text{through}~5~\text{has}~\text{positive}~\text{frequency}\]

No interior tick has zero dots, so there is no gap from 1 to 5.

04

Check visual outliers

\[\text{the}~\text{outer}~\text{values}~1~\text{and}~5~\text{are}~\text{equally}~\text{spaced}~\text{from}~\text{the}~\text{center}~\text{and}~\text{part}~\text{of}~\text{the}~\text{symmetric}~\text{taper}\]

Neither endpoint is isolated from the rest of the distribution.

05

Report the full shape classification

\[\text{modality}=\text{unimodal}~\text{with}~\text{mode}3;~\text{symmetry}=\text{symmetric}/\text{mound}-\text{shaped}~\text{about}3;~\text{gaps}=\text{none}~\text{from}1–5;~\text{visual}~\text{outliers}=\text{none};~\text{evidence}=\text{counts}~\text{pair}~1↔5~\text{as}1,~2↔4~\text{as}2,~\text{with}~\text{peak}4~\text{at}3\]

This separately states modality, symmetry, gaps, outliers, and the exact frequency evidence.

+

Another way

  1. Fold the frequency sequence 1,2,4,2,1 across its center; the left and right entries pair exactly.

!

Common mistake

Do not label 5 an outlier merely because it is an endpoint. It mirrors value 1 and continues the symmetric frequency pattern.

Worked statistical record: modality=unimodal with mode3; symmetry=symmetric/mound-shaped about3; gaps=none from1–5; visual outliers=none; evidence=counts pair 1↔5 as1, 2↔4 as2, with peak4 at3
modality=unimodal with mode3; symmetry=symmetric/mound-shaped about3; gaps=none from1–5; visual outliers=none; evidence=counts pair 1↔5 as1, 2↔4 as2, with peak4 at3