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Data AnalysisCore

One-Variable Data: Distributions and Measures

Read histograms, box plots, and dot plots — and predict how outliers move the mean while the median stands still.

⏱ 7 min📍 Learn first: Percentages
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Learn One-Variable Data: Distributions and Measures in the format that works for you

Start with a quick audio review, take the deeper podcast lesson, or move directly into the visual guide and knowledge check below.

13 free resources
00:00 / 01:34

Range IQR and Standard Deviation Shortcuts

A short spoken refresher on this concept.

00:00 / 18:35

Why Range and Standard Deviation Matter

The full lesson, worked through in depth.

Concept Snapshot

Definition
Data on a single measurement, summarized by center (mean, median, mode) and spread (range, standard deviation).
What the SAT tests
Computing and comparing measures of center, reading distribution shapes, and reasoning about outlier effects.
Key rule
Mean uses every value (outlier-sensitive). Median uses position (outlier-resistant). Skew pulls the mean toward the tail.
Most common mistake
Taking the median of a frequency table's values without weighting by their frequencies.
Prerequisite
Percentages

Visual Explanation

What to picture: Mean uses every value (outlier-sensitive). Median uses position (outlier-resistant). Skew pulls the mean toward the tail.

A diagram for this concept is in production. Until it is ready, the worked example below shows the same relationship step by step.

Concept Mind Map

DefinitionData on a single measurement, summarized by center (mean, median, mode) and spread (range, standard deviation).
Key ruleMean uses every value (outlier-sensitive). Median uses position (outlier-resistant). Skew pulls the mean toward the tail.

One-Variable Data: Distributions and Measures

Data Analysis

Outlier EffectHow does removing the largest value change mean vs. median?
Compare DistributionsTwo dot plots — which has the greater mean or spread?
Frequency Table CenterFind the median from value-count pairs.

How the SAT Tests It

Pattern

Outlier Effect

How does removing the largest value change mean vs. median?

Clue: "if the outlier is removed…"
Pattern

Compare Distributions

Two dot plots — which has the greater mean or spread?

Clue: Side-by-side plots
Pattern

Frequency Table Center

Find the median from value-count pairs.

Clue: A table with a frequency column

Worked Example

Nine houses sell for prices with mean $300k and median $260k. The $900k outlier sale is removed. What happens to each measure?

Predict the direction each center measure moves when the extreme value leaves.
The mean absorbed the $900k directly. New mean: (9×300 − 900)/8 = 1800/8 = $225k — a big drop.
The median only cares about middle position. Dropping the top value shifts the middle slightly down at most — from 5th of 9 to between 4th and 5th of 8.
Mean falls sharply ($300k → $225k); median barely moves. Outliers grab the mean, not the median.

Try It Yourself

A data set has mean 50 and median 42. Which is the most plausible explanation for the difference?

Common Traps

What students do

Treating mean and median as interchangeable.

Why it fails

They answer different questions — the SAT tests whether you know WHICH one outliers can move.

Do this instead

Tag each measure: mean = arithmetic with every value; median = middle position. When extremes enter or leave, only the mean chases them.

Interactive Flashcards

Review deckCard 1 of 8

Four-Question Knowledge Check

4 quick questions drawn from this concept's own snapshot, patterns and traps.

Question 1 of 4. Which statement is the key rule for this concept?

Audio Lesson

Why Range and Standard Deviation Matter

Mean vs. median behavior across distribution shapes.

00:00 / 18:35

18:35 · the full lesson

What to notice

Mean uses every value (outlier-sensitive). Median uses position (outlier-resistant). Skew pulls the mean toward the tail.

What the lesson covers

  • 0:00 The two centers
  • 1:30 Outlier experiments
  • 3:00 Reading skew
  • 4:20 Practice

Study Guide and Transcript

One-page study guide

The snapshot, the key rule, the patterns and the traps, on one page.

Source-Grounded AI Tutor

Ask the One-Variable Data: Distributions and Measures notebook

A tutor grounded in this concept's own sources — ask for a simpler explanation, another example, or why a particular answer is wrong. It is in development and will appear here once its sources pass academic review.

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